Fields Institute Communications Volume 00, 0000

Two Step Descent in Modular Galois Theory, Theorems of Burnside and Cayley, and Hilbert’s Thirteenth Problem

Shreeram S. Abhyankar

Mathematics Department Purdue University West Lafayette, IN 47907 USA

                          ram@cs.purdue.edu

Abstract. We propound a Descent Principle by which previously constructed equations over GF(q^{n})(X) may be deformed to have incarnations over GF(q)(X) without changing their Galois groups, where q = p^{u} > 1 is a power of a prime p and n is a positive integer. Currently this is achieved by starting with a vectorial (= additive) q-polynomial of q-degree m with Galois group GL(m, q) where m is any positive integer and then, under suitable conditions, enlarging its Galois group to GL(m, q^{n}) by forming its generalized iterate relative to an auxiliary irreducible polynomial of degree n. So, alternatively, we may regard this as an Ascent Principle. Elsewhere we proved this when m is square-free with GCD(mnu, 2p) = 1 = GCD(m, n). There the proof was based on CT (= the Classification Theorem of Finite Simple Groups) in its incarnation of CPT (= the Classification of Projectively Transitive Permutation Groups, i.e., subgroups of GL acting transitively on nonzero vectors). Here, without using CT, we shall give a direct proof of it when m = n = 2. The basic corner-stone of CPT is a Theorem of Burnside which says that a two-transitive permutation group has a unique minimal normal subgroup, which is either elementary abelian or simple. Since it is not very easy to find a proof of this Theorem of Burnside in the modern literature, we shall include a self-contained proof of it. We shall also discuss the relationship of Burnside’s Theorem to Cayley’s Theorem, and apply it to Hilbert’s Thirteenth Problem. Finally we shall motivate the whole matter by ideas from elementary analytic geometry.

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

2000 Mathematics Subject Classification. Primary 12F10, 14H30, 20D06, 20E22.

Key words and phrases. Descent, Galois group, Iteration.

Abhyankar’s work was partly supported by NSF Grant DMS 97-32592 and NSA grant MDA 904-99-1-0019.

1 Introduction

Let q = p^{u} > 1 be a power of a prime p, let m > 0 and n > 0 be integers, and let GF(q) ⊂ k_{q} ⊂ K ⊂ Ω be fields where Ω is an algebraic closure of K and where, as usual, GF(q) denotes the Galois field of q elements. Also let E = E(Y) be a monic separable vectorial q-polynomial of q-degree m in Y over K, i.e.,

E = E(Y) = Y^{q^{m}} + ∑_{i=1}^{m} X_{i}Y^{q^{m−i}} with X_{i} ∈ K and X_{m} ≠ 0

where the elements X_{1}, …, X_{m} need not be algebraically independent over k_{q}. Now the set of all roots of E in Ω is an m-dimensional GF(q)-vector-subspace V of Ω and, since GF(q) is assumed to be a subfield of k_{q} and hence of K, every K-automorphism of the splitting field K(V) of E over K induces a GF(q)-linear transformation of V. Consequently Gal(E, K) < GL(V), i.e., the Galois group of E over K may be regarded as a subgroup of the group GL(V) of all nonsingular linear transformations of V (see [3]). If we do not assume GF (q) ⊂ k_{q} then we only get Gal(E, K) < ΓL(V), where ΓL(V) is the group of all semilinear transformations of V (see [7]). In [4] and [7] it is shown that if K = k_{q}(X_{1}, …, X_{m}) and sufficiently many of the X_{i} are algebraically independent over k_{q} with the remaining being zero, then Gal(E, K) = GL(V); this under the assumption that k_{q} contains GF(q), whereas if k_{q} coincides with the prime field GF(p) then Gal(E, K) = ΓL(V). By fixing a basis of V we may identify GL(V) with GL(m, q), and ΓL(V) with ΓL(m, q). Here GL(m, q) is the group of all nonsingular m by m matrices over GF(q), and ΓL(V) consists of all pairs (g, γ) where g is a member of GL(m, q) and γ is an automorphism of GF(q), the pairs being multiplied as in a semi-direct product.

To prevent the said bloating of the Galois group towards ΓL we take recourse of generalized iteration which we shall define in a moment. This gives rise to the descent from a Galois group being realizable over GF(q) to its realizability over GF(p). Alternatively, we may call it ascent because, as we shall see, it enlarges a Galois group from GL(m, q) to GL(m, q^{n}). For the history of how Galois theory of q-polynomials came to play a significant role in the calculation of algebraic fundamental groups of affine curves, see papers [1] and [2]; for the higher dimensional story, see paper [5].

Definition 1.1 (Generalized Iteration) For every nonnegative integer j we inductively define the (ordinary) j-th iterate E^{[[j]]} of E by putting E^{[[0]]} = E^{[[0]]}(Y) = Y, E^{[[1]]} = E^{[[1]]}(Y) = E(Y), and E^{[[j]]} = E^{[[j]]}(Y) = E(E^{[[j−1]]}(Y)) for all j > 1. Next we define the generalized r-th iterate E^{[r]} of E for any r = r(T) = ∑ r_{i}T^{i} ∈ Ω[T] with r_{i} ∈ Ω (and r_{i} = 0 for all except a finite number of i), where T is an indeterminate, by putting E^{[r]} = E^{[r]}(Y) = ∑ r_{i}E^{[[i]]}(Y). Note that, for the Y -derivative E^{[r]}_{Y} (Y) of E^{[r]}(Y) we clearly have

E^{[r]}_{Y} (Y) = E^{[r]}_{Y} (0) = r(X_{m})  (1.2)

and hence if r(X_{m}) ≠ 0 then E^{[r]} is a separable vectorial q-polynomial over Ω whose q-degree in Y equals m times the T -degree of r. Also note that the definition of E^{[r]} remains valid for any vectorial E without assuming it to be monic or separable. Moreover, in such a general set-up, this makes the additive group of all vectorial q-polynomials E = E(Y) in Y over Ω into an Ω[T]-premodule having all the properties of a module except the left distributive law and the associativity of multiplication, i.e., for all r, r′ ∈ Ω[T] we have E^{[r+r′]} = E^{[r]} + E^{[r′]}, but for all E, E′ over Ω we need not have (E + E′)^{[r]} = E^{[r]} + E′^{[r]}, and in general E^{[rr′]} need

not be equal to (E^{[r]})^{[r′]}. At any rate, E^{[[j]]} of the previous notation corresponds to E^{[T^{j}]} in the present notation. Reverting to the fixed monic separable vectorial E exhibited in (1.1), the said premodule structure makes Ω into a GF(q)[T]-module when for every r ∈ GF(q)[T] and z ∈ Ω we define the “product” of r and z to be E^{[r]}(z); we denote this GF(q)[T]-module by Ω_{E}, and we call this the iteration module of E on Ω. Now let us fix

s = s(T) ∈ R = GF(q)[T] of T -degree n with s(X_{m}) ≠ 0  (1.3)

and note that then E^{[s]} is a separable vectorial q-polynomial of q-degree mn in Y over K. Let V^{[s]} be the set of all roots of E^{[s]} in Ω, and note that then V^{[s]} is an (mn)-dimensional GF(q)-vector-subspace of Ω. Let GF(q, s) = R/sR where sR is the ideal generated by s in R = GF(q)[T], and let ω : R → GF(q, s) be the canonical epimorphism. Now V^{[s]} is a submodule of Ω_{E} and as such it is annihilated by sR and hence we may regard it as a GF(q, s)-module, and we call it the iteration module of (E, q) at s; note that then, for every r ∈ R and z ∈ Ω, the “product” of ω(r) and z is given by ω(r)z = E^{[r]}(z) = ∑ r_{i}E^{[i]}(z), and for every g ∈ Gal(K(V^{[s]}), K) we have g(ω(r)z) = ∑ g(r_{i})E^{[i]}(g(z)) = (ω(r))g(z); also note that for all r ∈ R and z ∈ Ω we have rz = ω(r)z = E^{[r]}(z) = θ(r, z) with θ(r, z) ∈ (GF(q)[X_{1}, …, X_{m}])[z]. It follows that, in a natural manner,

Gal(E^{[s]}, K) < GL(V^{[s]})  (1.4)

where GL(V^{[s]}) is the group of all GF(q, s)-linear automorphisms of V^{[s]}, by which we mean all additive isomorphisms σ : V^{[s]} → V^{[s]} such that for all η ∈ GF(q, s) and z ∈ V^{[s]} we have σ(ηz) = ησ(z). Note that

s irreducible in R ⇒ GL(V^{[s]}) ≈ GL(m, q^{n})  (1.5)

where ≈ denotes isomorphism.

Definition 1.6 (Generic Polynomials) When we want to assume that, for a sequence of integers J^{∗} = (J_{1}^{∗}, …, J_{W^{∗}}^{∗}) with 1 ≤ J_{1}^{∗} < ⋯ < J_{W^{∗}}^{∗} ≤ m, the elements X_{J_{1}^{∗}}, …, X_{J_{W^{∗}}^{∗}} are algebraically independent over k_{q} and K = k_{q}(X_{J_{1}^{∗}}, …, X_{J_{W^{∗}}^{∗}}) with X_{i} = 0 for all i ∉ {J_{1}^{∗}, …, J_{W^{∗}}^{∗}}, we may express this by saying that we are in the generic case of type J^{∗}, and we may indicate it by writing E_{m,q}^{∗} for E and K^{∗} for K; moreover we put R^{∗} = k_{q}[X_{J_{1}^{∗}}, …, X_{J_{W^{∗}}^{∗}}]. When J^{∗} is the complete sequence J^{♯} = (1, …, m), we may say that we are in the very generic case to mean that we are in the generic case of type J^{♯}, and we may indicate this by writing E_{m,q}^{♯} for E and K^{♯} for K; moreover we put R^{♯} = k_{q}[X_{1}, …, X_{m}]. When J^{∗} is the singleton J^{♭} = (m) we may say that we are in the binomial case to mean that we are in the generic case of type J^{♭}, and we may indicate this by writing E_{m,q}^{♭} for E and K^{♭} for K; moreover we put R^{♭} = k_{q}[X_{m}]. When J^{∗} is the pair J^{†} = (m − 1, m) with m > 1 we may say that we are in the trinomial case to mean that we are in the generic case of type J^{†}, and we may indicate this by writing E_{m,q}^{†} for E and K^{†} for K; moreover we put R^{†} = k_{q}[X_{m−1}, X_{m}]. Finally, when J^{∗} is the sequence J^{‡} = (m − ν_{1}, …, m − ν_{W} , m), where m = ν_{0} > ν_{1} > ⋯ > ν_{W} = 1 are all the positive divisors of m, we may say that we are in the divisorial case to mean that we are in the generic case of type J^{‡}, and we may indicate this by writing E_{m,q}^{‡} for E and K^{‡} for K; again we put R^{‡} = k_{q}[(X_{i})_{i∈J^{‡}}]; note that if m = 1 then we are in the divisorial case with W = 0. As usual, for any other sequence of integers J^{∗∗} = (J_{1}^{∗∗}, …, J_{W^{∗∗}}^{∗∗}) with 1 ≤ J_{1}^{∗∗} < ⋯ < J_{W^{∗∗}}^{∗∗} ≤ m, we write J^{∗∗} ⊂ J^{∗} or J^{∗} ⊃ J^{∗∗} to mean that the sequence J^{∗∗} is a subsequence of the sequence J^{∗}. In particular, we always have J^{∗} ⊂ J^{♯}. Likewise, if J^{†} ⊂ J^{∗} then we must have m > 1. Moreover, if m is prime then J^{†} = J^{‡}. Similarly, if m = 1 then J^{♭} = J^{‡} = J^{♯}. With this notation at hand, part of what was proved in [4] and [7] can be stated as:

Lemma 1.7 (Trinomial Lemma) If J^{†} ⊂ J^{∗} then in the generic case of type J^{∗} we have Gal(E_{m,q}^{∗}, K^{∗}) = GL(m, q).

The above Definition (1.1) is a paraphrase of Remark (3.30) of [7]. In [7, Note (3.37)] we posed the following problem about generalized iterations.

Problem Show that in the very generic case we have Gal(E^{♯[s]}, K^{♯}) = GL(V^{[s]}).

In [8] we proved this when s = T^{n} and in [7, Theorem (3.25)] we semilinearized that result. Likewise in [9] we proved it under the assumptions that s is irreducible and m is a square-free integer with GCD(m, n) = 1 and GCD(mnu, 2p) = 1; actually, what we proved in [9, (1.18)] was the following slightly more general result:

Theorem 1.8 (Weak Divisorial Theorem) Assume that s is irreducible in R, and J^{‡} ⊂ J^{∗}. Also assume that m is a square-free integer with GCD(m, n) = 1, and GCD(mnu, 2p) = 1. Then in the generic case of type J^{∗} we have Gal(E_{m,q}^{∗[s]}, K^{∗}) = GL(V^{[s]}) ≈ GL(m, q^{n}).

Now CPT (= the Classification of Projectively Transitive Permutation Groups, i.e., subgroups of GL acting transitively on nonzero vectors) is a remarkable consequence of CT (= the Classification Theorem of Finite Simple Groups). The implication CT ⇒ CPT was mostly proved by Hering [25], [26]; it is also discussed by Cameron [15], Kantor [32], and Liebeck [33]. The proof of (1.8) given in [9] makes essential use of the following weaker version of CPT.

Theorem 1.9 (Weak CPT) Let d be an odd positive integer, and let G < GL(d, p) be transitive on the nonzero vectors GF(p)^{d} \{0}. Then there exist positive integers b, c with bc = d and a group G_{0} with SL(b, p^{c}) < G_{0} < ΓL(b, p^{c}) such that G ≈ G_{0}.

The m = 1 case of (1.8), without the hypothesis GCD(mnu, 2p) = 1, was proved by Carlitz [17] (also see Hayes [24]) in connection with his explicit class field theory. In our proof of (1.8) we used the following variation of Carlitz’s result which we reproved as Theorem (1.20) in [9]; recall that a univariate polynomial F̃(Y) = ∑_{i=0}^{N} F̃_{i}Y^{i} of positive degree N in Y is said to be Eisenstein relative to (R̃, M̃), where M̃ is a prime ideal in a ring R̃, if F̃_{N} ∈ R̃ \ M̃, F̃_{i} ∈ M̃ for 1 ≤ i ≤ N − 1, and F̃_{0} ∈ M̃ \ M̃^{2}.

Lemma 1.10 (Carlitz Irreducibility Lemma) Assume that s is irreducible in R, and J^{♭} ⊂ J^{∗}. Let s^{∗}(T) be a nonconstant irreducible factor of s(T) in k_{q}[T], and let M^{∗} be the ideal in R^{∗} = k_{q}[X_{J_{1}^{∗}}, …, X_{J_{W^{∗}}^{∗}}] generated by X_{J_{1}^{∗}}, …, X_{J_{W^{∗}−1}^{∗}}, s^{∗}(X_{m}). Then, for m = 1, in the generic case of type J^{∗} we have that M^{∗} = s^{∗}(X_{m})R^{∗} is a maximal ideal in R^{∗} = k_{q}[X_{m}], Y^{−1}E_{1,q}^{∗[s]}(Y) is Eisenstein relative to (R^{∗}, M^{∗}), Y^{−1}E_{1,q}^{∗[s]}(Y) is irreducible in K^{∗}[Y], and Gal(E_{1,q}^{∗[s]}, K^{∗}) = GL(V^{[s]}) ≈ GL(1, q^{n}). Moreover, without assuming m = 1, but assuming GCD(m, n) = 1, in the generic case of type J^{∗} we have that M^{∗} is a maximal ideal in R^{∗}, Y^{−1}E_{m,q}^{∗[s]}(Y) is Eisenstein relative to (R^{∗}, M^{∗}), Y^{−1}E_{m,q}^{∗[s]}(Y) is irreducible in K^{∗}[Y], and Gal(E_{m,q}^{∗[s]}, K^{∗}) has an element of order q^{mn} − 1.

The following number-theoretic result was originally proved in Zsigmondy’s 1892 paper [37]; proofs can also be found in Birkhoff-Vandiver’s 1904 paper [14], Dickson’s 1905 paper [20], Artin’s 1955 paper [11], and Feit’s 1988 paper [22].

Theorem 1.11 (Zsigmondy’s Theorem) Let M > 1 and N > 1 be any integers. Assume that (M, N) ≠ (2, 6). Also assume that (M, N) ≠ (2^{i} − 1, 2) for any integer i > 1 (recall that a prime number of the form 2^{i} −1 is called a Mersenne prime, and in that case i is automatically a prime). Then M^{N} − 1 has a prime divisor which does not divide M^{N′} − 1 for any positive integer N′ < N.

The following consequence of (1.11) is another auxiliary result which we used in the proof of (1.8); in [9] we proved it as Lemma (1.22).

Lemma 1.12 (Order Divisibility Lemma) Let b, c, l be any positive integers with b odd such that bc = ml and |SL(b, p^{c})| divides |GL(m, p^{l})|. Then b divides m.

In proving (1.8), in addition to items (1.9), (1.10) and (1.12), we also used the first part of the following well-known versatile Lemma which was initiated by Singer [35] (cf. pages 34 and 105 of Dembowski [19], pages 84–91 of Lidl-Niederreiter [34], and pages 187–189 of Huppert [29]) and which was stated as Lemma (1.23) in [9].

Lemma 1.13 (Singer Cycle Lemma) Let A ∈ GL(m, q) have order e = q^{m} − 1. Then det(A) has order ϵ = q − 1, and A acts transitively on the nonzero vectors GF(q)^{m} \ {0}, i.e., it is an e-cycle in the symmetric group S_{e} (and as such it is called a Singer Cycle). Moreover, in GL(m, q) all subgroups generated by such elements, i.e., all cyclic subgroups of order e, form a nonempty complete set of conjugates.

In the present paper we shall settle another case of the above Problem by proving the following:

Theorem 1.14 (Two Step Theorem) Assume that s is irreducible in R with m = n = 2. Then in the trinomial case we have Gal(E_{m,q}^{†[s]}, K^{†}) = GL(V^{[s]}) ≈ GL(m, q^{n}).

In Section 4 we shall deduce this from the following two Lemmas (1.15) and (1.16) which we shall prove in Sections 2 and 3 respectively. Specifically, for (1.15.1) see (2.11) to (2.14), for (1.15.2) see (2.15), for (1.15.3) see (2.16), and for (1.15.4) see (2.17). Note that in (1.16), the “Packet” refers to a Packet (= Bunch) of Roots.

Lemma 1.15 (Trinomial Computation Lemma) Assume that s = s(T) ∈ R = GF(q)[T] is monic irreducible of degree n = 2 and assume that we are in the trinomial case. Let us put E(Y) = E_{m,q}^{†}(Y) = Y^{q^{m}} +XY^{q} +ZY and R^{†} = k_{q}[X, Z] with X = X_{m−1} and Z = X_{m}, and let s_{1}^{†}(Z), …, s_{e}^{†}(Z) be the distinct nonconstant monic irreducible factors of s(Z) in R_{Z}^{†} = k_{q}[Z] (where e = 1 or 2). Then:

1.15.1 s(Z) = ∏_{1≤i≤e} s_{i}^{†}(Z) and E^{[s]}(Y) = ∑_{0≤i≤2m} A_{i}Y^{q^{i}} where A_{i} ∈ R_{0}^{†} = GF(q)[X, Z] for 0 ≤ i ≤ 2m with A_{2m} = 1, A_{1} ∈ Xs(Z)R_{0}^{†}, A_{0} = s(Z), and upon letting \overline{M} = ∪_{1≤i≤e}\overline{M}_{i} where \overline{M}_{i} is the the set of all maximal ideals M^{†} in R^{†} such that (A_{0}, …, A_{2m−1})R^{†} ⊂ M^{†} and M^{†} ∩ R_{Z}^{†} = s_{i}^{†}(Z)R_{Z}^{†} we have that: the localization R_{M^{†}}^{†} of R^{†} at any member M^{†} of \overline{M} is a two dimensional regular local domain; \overline{M} is exactly the set of all prime ideals M^{†} in R^{†} with (A_{0}, …, A_{2m−1})R^{†} ⊂ M^{†}; and \overline{M} is also exactly the set of all prime ideals M^{†} in R^{†} such that Y^{−1}E^{[s]}(Y) is Eisenstein relative to (R^{†}, M^{†}). Moreover we have the following:

1.15.2 If m is odd then for 1 ≤ i ≤ e we have \overline{M}_{i} = \{M_{i}^{†}\} where M_{i}^{†} = (X, s_{i}^{†}(Z))R^{†}.

1.15.3 If m is even and m > 2 then for 1 ≤ i ≤ e we have \overline{M}_{i} = ∅.

1.15.4 If m = 2 then for 1 ≤ i ≤ e we have 1 ≤ |\overline{M}_{i}| ≤ q + 1, and for any M^{†} ∈ \overline{M} we have A_{2} ∉ M^{†2} + A_{0}R^{†}.

Lemma 1.16 (Packet Throwing Lemma) Let M̃ be the maximal ideal in a regular local domain R̃ of dimension d > 0 with quotient field K̃. Let F̃(Y) = ∑_{0≤i≤N} F̃_{i}Y^{i} be a polynomial of degree N > 0 in Y which is Eisenstein relative to (R̃, M̃). [Note that then for some elements F_{2}, …, F_{d} in R̃ we have (F̃_{0}, F_{2}, …, F_{d})R̃ = M̃.] Let K̂ = K̃(η) where η is an element in an overfield of K̃ with F̃(η) = 0, and let R̂ = R̃[η] and M̂ = ηR̂ + M̃R̂. Then R̂ is the integral closure of R̃ in K̂, R̂ is a d dimensional regular local domain with maximal ideal M̂, M̂ ∩ R̃ = M̃, and for any η̂ ∈ K̂ with F̃(η̂) = 0 and any F_{2}, …, F_{d} in R̃ with (F̃_{0}, F_{2}, …, F_{d})R̃ = M̃ we have (η̂, F_{2}, …, F_{d})R̂ = M̂, and hence for any η̂ ∈ K̂ with F̃(η̂) = 0 we have η̂ ∈ M̂ \ M̂^{2}. Moreover, if for some positive integer D < N − 1 we have F̃_{D} ∉ M̃^{2} + F̃_{0}R̃ and F̃_{i} ∈ M̃^{D+2−i} + F̃_{0}R̃ for 1 ≤ i ≤ D − 1, and η_{1}, …, η_{D} are pairwise distinct elements in K̂ with F̃(η_{j}) = 0 for 1 ≤ j ≤ D, then F̃(Y) = F̂(Y) ∏_{1≤j≤D}(Y − η_{j}) where F̂(Y) is a polynomial of degree N − D in Y which is Eisenstein relative to (R̂, M̂).

It may be noted that CPT constitutes half of CDT = the Classification of Doubly Transitive Permutation Groups, which was the first amazing consequence of CT; see Cameron [15] and Kantor [32]. Now the basic foundation of CDT is Burnside’s Theorem, which he proved in the first 1897 edition of his famous book [13] on Groups of Finite Order. This Theorem of Burnside says that a two-transitive permutation group has a unique minimal normal subgroup, which is either elementary abelian or nonabelian simple. In the first case, which corresponds to CPT, the minimal normal subgroup is isomorphic to the additive group of the d dimensional vector space GF(π)^{d} over the prime field GF(π), where d is some positive integer and π is some prime, and the given two-transitive group can be viewed as a subgroup of the affine general linear group AGL(d, π). Note that AGL(d, π) is the semidirect product of GF(π)^{d} and GL(d, π), i.e., GF(π)^{d} ◁ AGL(d, π) and GL(d, q) < AGL(d, π) with GF(π)^{d} ∩ GL(d, π) = 1, where ◁ denotes normal subgroup, and AGL(d, π) is generated by its subgroups GF(π)^{d} and GL(d, π).

It is not very easy to find a proof of Burnside’s Theorem in the modern literature. Even if you do, it is likely to use Frobenius’ Theorems, as in Wielandt [36, p. 32, 12.4] or Huppert-Blackburn [30, Vol. III, p. 233, 7.12]. Frobenius’ Theorem, which he proved in his 1901 paper [23], says that in a finite transitive permutation group in which only the identity fixes two points, the set of all fixed point free elements, together with identity, form a subgroup. As I said in Footnotes 37 to 40 on page 89 of my 1992 Bull AMS paper [2], there is no proof of Frobenius’ Theorem without using character theory, i.e., traces of complex matrices in a homomorphic image of the group. Again as said on page 89 of [2], in the second 1911 edition of his book [13], Burnside gives two proofs of his theorem, one using Frobenius’ Theorem and the other without. As a public service, which really means for my own instruction, in Section 5 (see Theorem 5.15), I shall give a self-contained proof of Burnside’s Theorem without using Frobenius’ Theorem. In Section 6, I shall give a general perspective on Burnside’s Theorem in terms of basic analytic geometry; the reader may wish to read the motivational Section 6 before embarking on the rather technical Section 5. In Section 7, I shall show how this relates to Cayley’s Theorem which says that every finite group is isomorphic to a permutation group. In Section 8, I shall make some further remarks pertaining to Burnside’s Theorem. In Section 9, I shall end up by applying Burnside’s Theorm to Hilbert’s Thirteenth Problem.

In a forthcoming paper I shall show that the Weak Divisorial Theorem (1.8) remains true when we replace the assumption that GCD(mnu, 2p) = 1 by the less restrictive assumption that GCD(mnu, p) = 1.

2 Trinomial Calculations

Suppose we are in the trinomial case, and let us put R_{0}^{†} = GF(q)[X, Z] and R^{†} = k_{q}[X, Z] with X = X_{m−1} and Z = X_{m}. Also let us put E = E_{m,q}^{†} and assume that s = s(T) ∈ R = GF(q)[T] is monic irreducible of degree n = 2. Then

E = E(Y) = Y^{q^{m}} + XY^{q} + ZY ∈ R_{0}^{†}[Y]

and

s(T) = T^{2} + aT + b ∈ R = GF(q)[T] with a, b in GF(q).

We want to discuss whether the irreducibility of s in R implies the irreducibility of Y^{−1}E^{[s]}(Y) in R^{†}[Y], and then we shall use the irreducibility of Y^{−1}E^{[s]}(Y) in R^{†}[Y] to calculate Gal(E^{[s]}, K^{†}) where K^{†} = k_{q}(X, Z). Recall that for r(T) = ∑ r_{i}T^{i} ∈ R the generalized r-th iterate of E is defined by putting E^{[r]} = E^{[r]}(Y) = ∑ r_{i}E^{[[i]]}(Y) where the ordinary i-th iterate of E is given by E^{[[i]]}(Y) = E(E(… E(Y) …)) with E occurring i-times, and with the understanding that E^{[[1]]}(Y) = E(Y) and E^{[[0]]}(Y) = Y. To find E^{[s]} we first calculate E^{[[2]]} thus:

E^{[[2]]}(Y) =  (E(E(Y))
=  E(Y)^{q^{m}} + XE(Y)^{q} + ZE(Y)
=  (Y^{q^{2m}} + X^{q^{m}}Y^{q^{m+1}} + Z^{q^{m}}Y^{q^{m}})
+X(Y^{q^{m+1}} + X^{q}Y^{q^{2}} + Z^{q}Y^{q}) + Z(Y^{q^{m}} + XY^{q} + ZY)
=  Y^{q^{2m}} + (X^{q^{m}} + X)Y^{q^{m+1}} + (Z^{q^{m}} + Z)Y^{q^{m}}
+X^{q+1}Y^{q^{2}} + X(Z^{q} + Z)Y^{q} + Z^{2}Y.

This gives us:

\begin{aligned}E^{[s]}(Y) = & E^{[[2]]}(Y) + aE(Y) + bY \\ = & Y^{q^{2m}} + (X^{q^{m}} + X)Y^{q^{m+1}} \\ & + (Z^{q^{m}} + Z + a)Y^{q^{m}} + X^{q+1}Y^{q^{2}} \\ & + X(Z^{q} + Z + a)Y^{q} + (Z^{2} + aZ + b)Y. \end{aligned}  (2.1)

Now E^{[s]}(Y) is divisible by Y, and so by its irreducibility we obviously mean the irreducibility of Y^{−1}E^{[s]}(Y). To this end, assuming s(T) to be irreducible in R, we want to examine whether E^{[s]}(Y) ≡ Y^{q^{2m}}mod some maximal ideal M in R_{0}^{†}, i.e., whether all the coefficients of E^{[s]}(Y), except the leading coefficient, belong to M. So let M be such. Then s(Z) = (the coefficient of Y in E^{[s]}) ∈ M ∩ R_{Z} with R_{Z} = GF(q)[Z]; since s(Z) is irreducible in R_{Z}, we must have M ∩ R_{Z} = s(Z)R_{Z}.

Since two coefficients of E^{[s]} have a piece of the form

f_{d}(Z) = Z^{q^{d}} + Z + a

let us investigate for which positive integers d is f_{d}(Z) divisible by s(Z). So let z be a root of s(Z) and note that then z is a primitive element of GF(q^{2}) over GF(q). Since s(Z) is irreducible over GF(q) we also see that, if q is even then a ≠ 0 and hence 2z + a = a ≠ 0, whereas if q is odd then 2z + a is a primitive element of GF(q^{2}) over GF(q) and hence again 2z + a ≠ 0; thus we always have 2z + a ≠ 0; by the shape of s(Z) we also see that −a is the trace of z relative to the field extension GF(q^{2})/ GF(q). Now ζ ↦ ζ^{q} is the only nonidentity automorphism of GF(q^{2}) over GF(q), and its square is the identity. Therefore, if d is even then z^{q^{d}} = z and hence f_{d}(z) = 2z+a ≠ 0, whereas if d is odd then z^{q^{d}} = z^{q} and hence f_{d}(z) = (z^{q}+z)+a = (the trace of z relative to the field extension GF(q^{2})/GF(q)) + a = −a + a = 0. Also note that the Z-derivative of f_{d}(Z) is 1 and hence f_{d}(Z) is devoid of multiple roots. Thus

\begin{cases}if d is odd then f_{d}(Z) = s(Z)g_{d}(Z) where \\ g_{d}(Z) ∈ R_{Z} is coprime to s(Z) and f_{d}(z) = 0 \end{cases}  (2.2)

whereas

\begin{cases}if d is even then f_{d}(Z) = s(Z)g′_{d}(Z) + 2Z + a with \\ g′_{d}(Z) ∈ R_{Z} and f_{d}(z) = 2z + a ≠ 0. \end{cases}  (2.3)

In the expression of E^{[s]} given in (2.1), if m > 2 then all the exhibited powers of Y are distinct, whereas if m = 2 then Y^{q^{m}} = Y^{q^{2}} and hence the middle two powers coincide. In particular, if m > 2 then X^{q+1} being the coefficient of Y^{q^{2}} must belong to M, and hence X must belong to M. Therefore by (2.2) we see that

\begin{cases}if m is odd then M = (X, s(Z))R_{0}^{†} is the unique maximal ideal in R_{0}^{†} \\ for which E^{[s]}(Y) ≡ Y^{q^{2m}} mod M \end{cases} (2.4)

and by (2.3) we see that

\begin{cases}if m is even with m > 2 then there is no maximal ideal M in R_{0}^{†} \\ for which E^{[s]}(Y) ≡ Y^{q^{2m}} mod M. \end{cases}  (2.5)

If d is even and q is even then by (2.3) we get 0 ≠ f_{d}(z) ∈ GF(q) and hence f_{d}(z)^{q−1} = 1 = −1; likewise, if d is even and q is odd then by (2.3) we get f_{d}(z) ∉ GF(q) but 0 ≠ f_{d}(z)^{2} = (2z + a)^{2} = 4s(z) + a^{2} − 4b = a^{2} − 4b ∈ GF(q), and hence f_{d}(z)^{q−1} ≠ 1 but (f_{d}(z)^{q−1})^{2} = (f_{d}(z)^{2})^{q−1} = 1, and therefore f_{d}(z)^{q−1} = −1. Thus

if d is even then f_{d}(z)^{q−1} = −1.  (2.6)

Let R_{X} = GF(q^{2})[X] with GF(q^{2}) = GF(q)[z], and let E_{X}(Y) be the polynomial obtained by substituting z for Z in E^{[s]}(Y). Applying (2.2) with d = 1, by (2.1) we see that

the coefficent of Y^{q} in E^{[s]}(Y) belongs to Xs(Z)R_{0}^{†}  (2.7)

and hence by (2.1) we get

E_{X}(Y) = Y^{q^{2m}} + (X^{q^{m}} + X)Y^{q^{m+1}} + f_{m}(z)Y^{q^{m}} + X^{q+1}Y^{q^{2}}  (2.8)

and therefore we have

if m = 2 then E_{X}(Y) = Y^{q^{4}} + (X^{q^{2}} + X)Y^{q^{3}} + (X^{q+1} + f_{2}(z))Y^{q^{2}}.  (2.9)

If m = 2 then by (2.3) we see that X^{q+1} + f_{2}(z) = ∏_{1≤i≤q+1}(X + x_{i}) where x_{1}, …, x_{q+1} are distinct elements in the splitting field k of X^{q+1}+f_{2}(z) over GF(q^{2}) and, for 1 ≤ i ≤ q + 1, by (2.6) we see that X + x_{i} divides X^{q^{2}−1} + 1 in k[X], and hence by (2.9) we see that E_{X}(Y) ≡ Y^{q^{4}} mod (X + x_{i})k[X], and therefore E^{[s]}(Y) ≡ Y^{q^{4}} mod M for some maximal ideal M in R_{0}^{†}. Thus

\begin{cases}if m = 2 then X^{q+1} + f_{2}(z) = ∏_{1≤i≤q+1}(X + x_{i}) \\ where x_{1}, …, x_{q+1} are distinct elements \\ in the splitting field k of X^{q+1} + f_{2}(z) over GF(q^{2}), \\ and for 1 ≤ i ≤ q + 1 we have E_{X}(Y) ≡ Y^{q^{4}} mod (X + x_{i})k[X], \\ and for some maximal ideal M in R_{0}^{†} we have E^{[s]}(Y) ≡ Y^{q^{4}} mod M. \end{cases} (2.10)

Let s_{1}^{†}(Z), …, s_{e}^{†}(Z) be the distinct nonconstant monic irreducible factors of s(Z) in R_{Z}^{†} = k_{q}[Z]. Since s(Z) is irreducible in GF(q)[Z] and GF(q) is perfect, we see that e = 1 or 2, and we have

s(Z) = ∏_{1≤i≤e} s_{i}^{†}(Z).  (2.11)

By (2.1) and (2.7) we also see that

E^{[s]}(Y) = ∑_{0≤i≤2m} A_{i}Y^{q^{i}} where A_{i} ∈ R_{0}^{†} for 0 ≤ i ≤ 2m  (2.12)

with

A_{2m} = 1 and A_{1} ∈ Xs(Z)R_{0}^{†} and A_{0} = s(Z).  (2.13)

Let \overline{M} = ∪_{1≤i≤e}\overline{M}_{i} where \overline{M}_{i} is the the set of all maximal ideals M^{†} in R^{†} such that (A_{0}, …, A_{2m−1})R^{†} ⊂ M^{†} and M^{†} ∩ R_{Z}^{†} = s_{i}^{†}(Z)R_{Z}^{†}. By (2.1) we have A_{m+1} = X^{q^{m}}+X and A_{0} = s(Z); from this it follows that if M^{†} is any prime ideal in R^{†} containing A_{m+1} and A_{0}, then M^{†} is a maximal ideal in R^{†}, the localization R_{M^{†}}^{†} of R^{†} at M^{†} is a two dimensional regular local domain, and M^{†} ∩ R_{Z}^{†} = s_{i}^{†}(Z)R_{Z}^{†} for some i ∈ {1, …, e}. Therefore in view of (2.11) to (2.13) we see that

\begin{cases}the localization R_{M^{†}}^{†} of R^{†} at any member M^{†} of \overline{M} \\ is a two dimensional regular local domain, \\ \overline{M} is exactly the set of all prime ideals M^{†} in R^{†} \\ with (A_{0}, …, A_{2m−1})R^{†} ⊂ M^{†}, and \\ \overline{M} is also exactly the set of all prime ideals M^{†} in R^{†} \\ such that Y^{−1}E^{[s]}(Y) is Eisenstein relative to (R^{†}, M^{†}). \end{cases}  (2.14)

Now by (2.4) we see that

\begin{cases}If m is odd then for 1 ≤ i ≤ e we have \overline{M}_{i} = \{M_{i}^{†}\} \\ where M_{i}^{†} = (X, s_{i}^{†}(Z))R^{†} \end{cases}  (2.15)

and by (2.5) we see that

If m is even and m > 2 then for 1 ≤ i ≤ e we have \overline{M}_{i} = ∅  (2.16)

and finally by (2.9) to (2.13) we see that

\begin{cases}If m = 2 then for 1 ≤ i ≤ e we have 1 ≤ |\overline{M}_{i}| ≤ q + 1, \\ and for any M^{†} ∈ \overline{M} we have A_{2} ∉ M^{†2} + A_{0}R^{†}. \end{cases}  (2.17)

3 Throwing Away a Packet of Roots

Let F̃(Y) = ∑_{0≤i≤N} F̃_{i}Y^{i} be a polynomial of degree N > 0 in Y which is Eisenstein relative to (R̃, M̃) where M̃ is the maximal ideal in the regular local domain R̃ of dimension d > 0. Note that then for some elements F_{2}, …, F_{d} in R̃ we have (F̃_{0}, F_{2}, …, F_{d})R̃ = M̃. Let K̂ = K̃(η) where K̃ is the quotient field of R̃, and η is an element in overfield of K̃ such that F̃(η) = 0. Also let R̂ = R̃[η] and M̂ = ηR̂ + M̃R̂. Then by the Generalized Eisenstein Criterion (i.e., the easily established fact which we used in Proposition 5.3 of [4] and Item 3.23 of [7]) we see that then R̂ is the integral closure of R̃ in K̂, R̂ is a d dimensional regular local domain with maximal ideal M̂, M̂ ∩ R̃ = M̃, and for any F_{2}, …, F_{d} in R̃ with (F̃_{0}, F_{2}, …, F_{d})R̃ = M̃ we have (η, F_{2}, …, F_{d})R̂ = M̂. From this it follows that for any η̂ ∈ K̂ with F̃(η̂) = 0 and any F_{2}, …, F_{d} in R̃ with (F̃_{0}, F_{2}, …, F_{d})R̃ = M̃ we have (η̂, F_{2}, …, F_{d})R̂ = M̂, and hence for any η̂ ∈ K̂ with F̃(η̂) = 0 we have η̂ ∈ M̂ \ M̂^{2}. Now assume that for some positive integer D < N − 1 we have F̃_{D} ∉ M̃^{2} + F̃_{0}R̃ and F̃_{i} ∈ M̃^{D+2−i} + F̃_{0}R̃ for 1 ≤ i ≤ D − 1, and assume that η_{1}, …, η_{D} are pairwise distinct elements in K̂ with F̃(η_{j}) = 0 for 1 ≤ j ≤ D. Then by what we have said above, we get η_{i} ∈ M̂ \ M̂^{2} for 1 ≤ i ≤ D, and obviously

F̃(Y) = F̂(Y) ∏_{1≤j≤D}(Y − η_{j})

where F̂(Y) ∈ K̂[Y] is of degree N − D and its leading coefficient is F̃_{N} which belongs to R̂\M̂. Since the domain R̂ is integrally closed in its quotient field K̂ (actually R̂ is a UFD), by Gauss’ Lemma we see that F̂(Y) ∈ R̂[Y]. By reducing both sides mod M̂, and remembering that F̃(Y) = CY^{N} mod M̂ with 0 ≠ C ∈ R̂/M̂, we see that F̂(Y) = CY^{N−D} mod M̂. Now F̃_{0} = F̃(0) = (−1)^{D}F̂(0) ∏_{1≤i≤D} η_{i} and hence if we show that F̃_{0} ∉ M̂^{D+2} then we would get F̂(0) ∉ M̂^{2} which would establish the Eisensteinness of F̂(Y) relative to (R̂, M̂). To do this, take F_{2}, …, F_{d} in R̃ with (F̃_{0}, F_{2}, …, F_{d})R̃ = M̃. Then by the condition on F̃_{D} we get F̃_{D} = AF̃_{0} + B_{2}F_{2} + ⋯ + B_{d}F_{d} where A, B_{2}, …, B_{d} are elements in R̃ such that B_{j} ∉ M̃ for some j. Likewise, by the condition on F̃_{i} for 1 ≤ i ≤ D − 1, we get ∑_{1≤i≤D−1} F̃_{i}η^{i} = A′ηF̃_{0} + C′ with A′ ∈ R̂ and C′ ∈ M̂^{D+2}. Since F̃_{i} ∈ M̃ for D < i < N, we also get ∑_{D<i≤N} F̃_{i}η^{i} ∈ M̃^{D+2}. Now the equation F̃(η) = 0 gives us A^{∗}F̃_{0} = (B_{2}F_{2} + ⋯ + B_{d}F_{d})η^{D} + C^{∗} where A^{∗} = −(1 + Aη^{D} + A′η) ∈ R̂ \ M̂ and C^{∗} = C′ + ∑_{D<i≤N} F̃_{i}η^{i} ∈ M̂^{D+2}; since (η, F_{2}, …, F_{d})R̂ = M̂ and B_{j} ∉ M̂ for some j, it follows that F̃_{0} ∉ M̂^{D+2}.

4 Two Step Theorem

Assume that m = 2 and s = s(T) ∈ R = GF(q)[T] is irreducible of degree n = 2. We want to show that then in the trinomial case we have Gal(E_{m,q}^{†[s]}, K^{†}) = GL(V^{[s]}) ≈ GL(m, q^{n}). Since multiplying s by a nonzero element of GF(q) amounts to multiplying E_{m,q}^{†[s]} by the same element, without loss of generality we may assume s to be monic. Let us put E(Y) = E_{m,q}^{†}(Y) = Y^{q^{m}} +XY^{q} +ZY and R^{†} = k_{q}[X, Z] with X = X_{m−1} and Z = X_{m}. Note that then K^{†} = k_{q}(X, Z), and by (1.4) and (1.5) we have Gal(E^{[s]}, K^{†}) < GL(V^{[s]}) ≈ GL(m, q^{n}) and hence it suffices to show that |Gal(E^{[s]}, K^{†})| ≥ |GL(m, q^{n})|. Since m = n = 2, we clearly have |GL(m, q^{n})| = (q^{4} − 1)(q^{4} − q^{2}), and hence it suffices to show that |Gal(E^{[s]}, K^{†})| ≥ (q^{4} − 1)(q^{4} − q^{2}). Let F̃(Y) = Y^{−1}E^{[s]}(Y) and N = q^{4} − 1. Then by (1.15.1) and (1.15.4) we see that F̃(Y) = Y^{N} + A_{3}Y^{q^{3}−1} + A_{2}Y^{q^{2}−1} + A_{1}Y^{q−1} + A_{0} with A_{0}, A_{1}, A_{2}, A_{3} in R^{†}, and there exists a maximal ideal M^{†} in R^{†} which contains the element A_{0}, A_{1}, A_{2}, A_{3}, and for any such maximal ideal M^{†} we have that the localization R̃ = R_{M^{†}}^{†} is a two dimensional regular local domain with maximal ideal M̃ = M^{†}R̃ and quotient field K̃ = K^{†}, the polynomial F̃(Y) is Eisenstein relative to (R̃, M̃), and for the elements A_{1} and A_{2} we have A_{1} ∈ A_{0}R̃ and A_{2} ∉ M̃^{2} + A_{0}R̃. Let R̂ = R̃[η] and K̂ = K̃(η) where η is an element in the algebraically closed overfield Ω of K̃ with F̃(η) = 0. By (1.16) we see that R̂ is the integral closure of R̃ in K̂, and R̂ is a two dimensional regular local domain with maximal ideal M̂ = ηR̂ + M̃R̂. Let R_{0} be the set of all nonzero members of R of T -degree less than n. Then, in the notation of (1.1), (ω(r)η)_{r∈R_{0}} are all the distinct “nonzero scalar multiples” of η in the (R/s)-vector space V^{[s]}, and clearly R_{0} is the set of all αT + β with (0, 0) ≠ (α, β) ∈ GF(q) × GF(q). This gives us q^{2} − 1 distinct roots of F̃(Y) in K̂. Therefore by taking D = q^{2} − 1 in (1.16) we see that F̃(Y) = F̂(Y) ∏_{r∈R_{0}}(Y −ω(r)η) where F̂(Y) is a polynomial of degree N −D = q^{4}−q^{2} in Y which is Eisenstein relative to (R̂, M̂). It follows that F̃(Y) and F̂(Y) are irreducible in K̃[Y] and K̂[Y] respectively. Therefore [K̂ : K̃] = q^{4} − 1 and [K̂′ : K̂] = q^{4} − q^{2} where we have taken a root η′ of F̂(Y) in Ω and put K̂′ = K̂(η′). Now clearly |Gal(E^{[s]}, K^{†})| ≥ [K̂ : K̃][K̂′ : K̂] and hence |Gal(E^{[s]}, K^{†})| ≥ (q^{4} − 1)(q^{4} − q^{2}).

5 Burnside’s Theorem

IN THE REST OF THE PAPER, the various letters p, q, m, n, X, Y, Z and so on, do NOT have the same meaning as in the previous Sections.

We want to consider a permutation group G on n letters, i.e., G < S_{n} where S_{n} is the symmetric group on n letters. It is sometimes better not to fix the letters to be the integers 1, …, n, but to let them be arbitrary. Recall that the symmetric group on a set W, i.e., the group of all bijections of W, is denoted by Sym(W). These days half the books compose maps from right to left, while the others do the opposite. We have been doing right to left, i.e., (ϕψ)(x) = ϕ(ψ(x)). At any rate, Sym(W) is a group under this composition.

The proof of Burnside’s Theorem is delicate. To state hypotheses precisely, it will help to have at hand several definitions. In the next six paragraphs we highlight those definitions that the reader might want to review later. These are the definitions of transitive, two-transitive, regular, and Frobenius, and also the definitions of semi-transitive, sesqui-transitive, semi-regular, and semi-Frobenius. Our terms semi-transitive and sesqui-transitive are similar to but not precisely the same as the more common 1/2-transitive and 3/2-transitive respectively. Much later, in the paragraph before Lemma (5.7), we shall introduce the definitons of primitive and imprimitive, and of the related notions of invariant partition, block, and system of imprimitivity.

So let G < S_{n} = Sym(W) with 1 ≤ |W | = n < ∞, i.e., let G be a permutation group on a nonempty finite set W of size n; we may refer to members of W as points. w ∈ W is a fixed point of g ∈ G means g(w) = w.

G is transitive means that any point of W can be mapped to any other point of W by an element of G. G is two-transitive means that n is at least two and any two points of W can be mapped to any other two points of W by an element of G. G is semi-regular means that only the identity of G has a fixed point. G is regular means that G is both transitive and semi-regular. Notice that if G is regular then |G| = n. Recall that Cayley’s Theorem states that every finite group is isomorphic to a permutation group, which is obviously regular, and in this manner we get all regular finite permutation groups. G is semi-Frobenius means that G is transitive, n is at least two, and only the identity of G has 2 fixed points. G is Frobenius means that G is semi-Frobenius but is not regular.

More generally, for a positive integer t, we say that G is t-transitive if t ≤ n and any t points of W can be mapped to any other t points of W by an element of G, and we say that G is t-antitransitive if t ≤ n and the identity is the only element of G having t fixed points. In the notation of [2], G is (t, τ) means G is t-transitive and τ-antitransitive, and then, regarding every G to be 0-transitive as well as 0antitransitive, we have (0, 1) = semi-regular, (1, 0) = transitive, (1, 1) = regular, (1, 2) = semi-Frobenius, (1, 2)\(1, 1) = Frobenius; a (t, t) group is said to be sharply t-transitive; a sharply 2-transitive group is also called sharp-Frobenius. For a finite field k, the one-dimensional affine general linear group AGL(1, k) consisting of all bijections of k of the form x ↦ ax+b with 0 ≠ a ∈ k and b ∈ k, is sharp-Frobenius; as stated on page 78 of [2], Zassenhaus’ Theorem says that the converse is true if we allow k to be a finite near field.

Given any g ∈ G we put u^{g} = g(u) for u ∈ W, and U^{g} = {u^{g} : u ∈ U} for U ⊂ W. For any u ∈ W we put G_{u} = {g ∈ G : u^{g} = u} = stab_{G}(u) = (stabilizer of u in G), and for any U ⊂ W we put G_{U} = {g ∈ G : u^{g} = u for all u ∈ U} = estab_{G}(U) = (elementwise stabilizer of U in G) < {g ∈ G : U^{g} = U} = stab_{G}(U) = (stabilizer of U in G). Given any H ⊂ G, for every u ∈ W we put u^{H} = {u^{h} : h ∈ H} = orb_{H}(u) = (H-orbit of u); by an H-orbit we mean the H-orbit of u for some u ∈ W. We call G semi-transitive if all the G-orbits have the same size, and we note the:

Lemma 5.1 (Obvious Lemma) If G is semi-transitive then, upon letting m to be the common size of all the G-orbits and μ to be their number, we have n = μm with m = |u^{G}| for all u ∈ W.

In the above type definitions, when necessary, reference to W may be made explicit. Thus, instead of saying that G is t-transitive we may say that G is ttransitive on W. If n ≥ 2 then, for any u ∈ W, G_{u} may be viewed as a subgroup of Sym(W \ {u}). We call G sesqui-transitive if G is transitive on W and, for every u ∈ W, G_{u} is semi-transitive on W \ {u}.

Given any u ∈ W, for g, h in G we have u^{g} = u^{h} ⇔ (h^{−1}g)(u) = u ⇔ h^{−1}g ∈ G_{u} ⇔ g and h belong to the same left coset of G_{u} in G. Thus u^{g} ↦ gG_{u} gives a bijection of u^{G} onto the set of all left cosets of G_{u} in G. Therefore:

Equation 5.2 (Orbit-Stabilizer Equation) |u^{G}| = |G|/|G_{u}| for all u ∈ W.

As an easy consequence of this we get the:

Lemma 5.3 (Order Lemma) We have the following:

5.3.1 G semi-transitive ⇒ |G_{u}| = |G_{v}| for all u, v in W.

5.3.2 G semi-regular ⇒ |u^{G}| = |G| for all u ∈ W.

5.3.3 G semi-regular ⇒ G semi-transitive and |G| divides n.

5.3.4 G transitive ⇒ |G| = n|G_{u}| for all u ∈ W.

5.3.5 G regular ⇔ G semi-regular and |G| = n ⇔ G transitive and |G| = n.

5.3.6 G semi-Frobenius ⇒ |G| divides n(n − 1).

5.3.7 G sesqui-transitive and |G| divides n^{2} ⇒ |G| = n.

Namely, (5.3.1) follows from (5.2) by noting that if G is semi-transitive then for all u, v in W we have |G_{u}| = |G|/m = |G_{v}| where m is the common size of all the G-orbits. (5.3.2) follows from (5.2) by noting that if G is semi-regular then G_{u} = 1 for all u ∈ W. (5.3.3) follows from (5.1) and (5.3.2). (5.3.4) follows from (5.2) by noting that if G is transitive then |u^{G}| = n for all u ∈ W. (5.3.5) follows from (5.3.2) and (5.3.4) by noting that G is transitive ⇔ |u^{G}| = n for all u ∈ W. (5.3.6) follows by noting that if G is semi-Frobenius then by (5.3.4) we get |G| = n|G_{u}| for all u ∈ W, and by taking (G_{u}, W \{u}) for (G, W) in (5.3.3) we see that |G_{u}| divides n − 1. Finally, to prove (5.3.7) assume that G is sesqui-transitive and |G| divides n^{2}; then fixing any u ∈ W, by (5.3.4) we have |G| = n|G_{u}| and hence |G_{u}| divides n and we want to show that |G_{u}| = 1, i.e., G_{u} = 1; now G_{u} is semi-transitive on W \{u}, and hence by taking (G_{u}, W \{u}) for (G, W) in (5.1) we get n−1 = μ′m′ where μ′ is the number of orbits of G_{u} on W \ {u} and m′ is their common size; by taking (G_{u}, W \ {u}) for (G, W) in (5.2) we also see that m′ divides |G_{u}| which we know divides n; thus m′ divides n as well as n − 1 and hence we must have m′ = 1, i.e., every orbit of G_{u} on W \ {u} has size 1; therefore G_{u} = 1.

For any g ∈ G we put h^{g} = ghg^{−1} for h ∈ G, and H^{g} = {h^{g} : h ∈ H} for H ⊂ G. Note that then for g, h in G, in the usual permutation notation we have

h = (\begin{matrix}u_{1} & … & u_{n} \\ v_{1} & … & v_{n} \end{matrix}) ⇒ h^{g} = (\begin{matrix}g(u_{1}) & … & g(u_{n}) \\ g(v_{1}) & … & g(v_{n}) \end{matrix}) ,

i.e., equivalently, we have:

Rule 5.4 (Conjugation Rule) h^{g}(g(u)) = g(h(u)) for all u ∈ W.

A bijection f : W → W^{♭} of sets induces the isomorphism f^{♯} : Sym(W) → Sym(W^{♭}) given by f^{♯}(h)(f(u)) = f(h(u)) for all h ∈ Sym(W) and u ∈ W; this may be called transportation isomorphism since it transports the group structure from Sym(W) to Sym(W^{♭}); in a sense, the above Conjugation Rule is also a transportation phenomenon; to bring out the analogy between these two, we note that in effect f^{♯}(h) = fhf^{−1}. In turn f^{♯} induces the (restriction) isomorphism f_{G}^{♯} : G → G^{♭} where G^{♭} = f^{♯}(G); we call f_{G}^{♯} a permutation-isomorphism, and we say that G and G^{♭} are permutation-isomorphic. If G is permutation-isomorphic to a permutation group G^{♭} then clearly: G is 2-transitive ⇔ G^{♭} is 2-transitive; G is regular ⇔ G^{♭} is regular; and so on. Upon taking W^{♭} = W and f ∈ G, by applying the above Conjugation Rule (with f replacing g) we get f^{♯}(h) = h^{f} for all h ∈ G, and hence for any u ∈ W we have f^{♯}(G_{u}) = G_{f(u)} and the induced isomorphism G_{u} → G_{f(u)} makes G_{u} permutation-isomorphic to G_{f(u)}. Consequently, we get the following:

Lemma 5.5 (Conjugacy Lemma) If G is transitive and W = {w_{1}, …, w_{n}}, then G_{w_{1}}, …, G_{w_{n}} are mutually permutation-isomorphic and they form a complete conjugacy class of subgroups of G. Moreover, the said permutation-isomorphism G_{w_{i}} → G_{w_{j}} is induced by a bijection W → W which sends w_{i} to w_{j}.

As a consequence of this we see that: G is two-transitive ⇔ G is transitive with n ≥ 2 and G_{u} is transitive on W \ {u} for some u ∈ W; G is regular ⇔ G is transitive and G_{u} = 1 for some u ∈ W; and so on. As another obvious fact we note that if G is regular then, by fixing any w ∈ W, we get a bijection f : W → G whose inverse is given by g ↦ g(w); the significance of this bijection f : W → G and the transportation isomorphism f^{♯} : Sym(W) → Sym(G) induced by it will be clear from the following Centralizer Lemma (5.6) and Normality Lemma (5.10).

Recall that for a group X, for any y ∈ X we have C_{X}(y) = (centralizer of y in X) = {x ∈ X : y^{x} = y}, and for any Y < X we have C_{X}(Y) = (centralizer of Y in X) = ∩_{y∈Y} C_{X}(y); moreover, y′ ∈ X centralizes Y means y′ ∈ C_{X}(Y), and Y′ ⊂ X centralizes Y means Y′ ⊂ C_{X}(Y); for instance: Y is abelian ⇔ Y centralizes Y. Assuming X to be finite, the left and right multiplication in X correspond to the monomorphisms L_{X} : X → Sym(X) and R_{X} : X → Sym(X) given by L_{X}(x)(y) = xy and R_{X}(x)(y) = yx^{−1} for all x, y in X. Their images L_{X}(X) and R_{X}(X) are clearly regular subgroups of Sym(X) and they may respectively be called the left and right regular permutation representations of X; also clearly L_{X}(X) and R_{X}(X) centralize each other and hence, as we shall see in a moment, they are the full centralizers of each other in Sym(X). Actually, this follows from the next lemma which we shall now prove.

Lemma 5.6 (Centralizer Lemma) For the centralizer G′ of G in Sym(W), we have the following:

5.6.1 If G is transitive then G′ is semi-regular. [It follows that if a transitive subgroup G of Sym(W) is centralized by a subgroup G′′ of Sym(W) then G′′ is semi-regular.]

5.6.2 If G and G′ are transitive then G and G′ are regular, G is the centralizer of G′ in Sym(W), and G′ (resp: G) is the only transitive subgroup of Sym(W) which centralizes G (resp: G′). [It follows that if a transitive subgroup G of Sym(W) is centralized by a transitive subgroup G′′ of Sym(W), then G and G′′ are the full centralizers of each other in Sym(W), and G′′ (resp: G) is the only transitive subgroup of Sym(W) which centralizes G (resp: G′′).]

5.6.3 If G is regular then G′ is also regular, and by fixing w ∈ W and letting f : W → G be the bijection whose inverse is given by g ↦ g(w), we have f^{♯}(G) = L_{G}(G) and f^{♯}(G′) = R_{G}(G) where f^{♯} : Sym(W) → Sym(G) is the isomorphism induced by f and L_{G}(G) and R_{G}(G) are the left and right permutation representations of G respectively, and hence G are G′ are permutation-isomorphic. [By (5.6.2) and (5.6.3) it follows that if G is regular then G′ (resp: G) is the unique transitive subgroup of Sym(W) which centralizes G (resp: G′).]

First, to prove (5.6.1), assume G transitive. If G′ is not semi-regular then we can find u ≠ v in W and g ∈ G′ with u^{g} = u and v^{g} ≠ v. Since G is transitive, we can find h ∈ G with h(u) = v. Then h^{g}(u^{g}) = v^{g} by the conjugation rule, and h^{g} = h because h and g commute. Thus v = h(u) = h^{g}(u^{g}) = v^{g} ≠ v which is a contradiction. Therefore G′ is semi-regular.

Next, to prove (5.6.2), assume that G and G′ are transitive. Then by (5.6.1) they must be regular, and hence |G| = n = |G′|. If G′′ < Sym(W) centralizes G then G′′ < G′, and if G′′ is also transitive then |G′′| ≥ n and hence G′′ = G′. Let G^{†} be the centralizer of G′ in Sym(W). Since G is transitive and centralizes G′, it follows that the G < G^{†} and G^{†} is transitive. Therefore by reversing the roles of G′ and G^{†} in what we have already proved, it follows that G = G^{†} and G is the only transitive subgroup of Sym(W) which centralizes G′.

Finally, to prove (5.6.3), assume that G is regular, and fixing w ∈ W, let f : W → G be the bijection whose inverse is given by g ↦ g(w), let f^{♯} : Sym(W) → Sym(G) be the isomorphism induced by f, and let L_{G}(G) and R_{G}(G) be the left and right permutation representations of G respectively. Then clearly f^{♯}(G) = L_{G}(G), and hence upon letting G^{‡} = f^{♯−1}(R_{G}(G)) we see that G^{‡} is a regular subgroup of Sym(W) which centralizes G and is permutation-isomorphic to it. By (5.6.2) we must have G^{‡} = G′.

Now we turn our attention to the concept of primitivity which, like sesquitransitivity, is between transitivity and two-transitivity. By a G-invariant partition (of W) we mean a partition T = (T_{i})_{1≤i≤r} of W into pairwise disjoint nonempty sets T_{1}, …, T_{r} (with W = T_{1} ∪ ⋯ ∪ T_{r}) of equal size |T_{i}| = s for 1 ≤ i ≤ r such that for every g ∈ G and i ∈ {1, …, r} we have g(T_{i}) = T_{j} for some j ∈ {1, …, r} which may or may not be equal to i; note that then r and s are positive integers with n = rs; we call (r, s) the type of T; the partition T is trivial means either r = 1 or s = 1. A G-system of imprimitivity is a nontrivial G-invariant partition. G is imprimitive if it is transitive and has a system of imprimitivity. G is primitive if it is transitive but not imprimitive. By a G-block we mean a nonempty subset U of W such that for every g ∈ G we have either U^{g} = U or U^{g} ∩ U = ∅; we call U a trivial block if either U = W or |U| = 1. For relating blocks with orbits and partitions, let us prove the:

Lemma 5.7 (Block Lemma) We have the following.

5.7.1 If u ∈ W and H < G then (u^{H})^{g} = (u^{g})^{H^{g}} for all g ∈ G.

5.7.2 If u ∈ W and G_{u} < H < G then u^{H} is a G-block with |u^{H}| = [H : G_{u}].

5.7.3 If U is a G-block and g, h are elements in G then U^{g} and U^{h} are G-blocks such that either U^{g} = U^{h} or U^{g} ∩ U^{h} = ∅.

5.7.4 If T = (T_{i})_{1≤i≤r} is a G-invariant partition then each T_{i} is a G-block.

5.7.5 If G is transitive and U is a G-block then, upon letting T = (T_{i})_{1≤i≤r} to be the family of all distinct sets of the form U^{g} as g varies over G (we then say that T is generated by U), we have that r = n/|U| and T is G-invariant partition of type (r, |U|), and hence in particular |U| divides n.

5.7.6 Conversely, if G is transitive and T = (T_{i})_{1≤i≤r} is a G-invariant partition then, for 1 ≤ i ≤ r, T_{i} is a G-block which generates T. Moreover, if G is transitive then: a G-block is trivial ⇔ the partition it generates is trivial.

5.7.7 If G is transitive and N ◁ G then the set of all N-orbits is a G-invariant partition, and hence N is semi-transitive and the size of any N-orbit divides n.

To prove (5.7.1) let u ∈ W and H < G. Then for any g ∈ G, we have h(u)^{g} = h^{g}(u^{g}) for every h ∈ H, and hence (u^{H})^{g} = (u^{g})^{H^{g}}.

To prove (5.7.2) let u ∈ W and G_{u} < H < G. Then for any g ∈ G we have: (u^{H})^{g} ∩ u^{H} ≠ ∅ ⇒ g(f(u)) = h(u) for some f, h ∈ H ⇒ h^{−1}gf ∈ G_{u} ⊂ H ⇒ g ∈ H ⇒ (u^{H})^{g} = u^{H}. Therefore u^{H} is a G-block. Moreover, for any f, h ∈ H we have f(u) = h(u) ⇔ h^{−1}f ∈ G_{u}, and hence |u^{H}| = [H : G_{u}].

To prove (5.7.3) let U be a G-block and let g, h be elements in G. Then for any f ∈ G we have: (U^{g})^{f} ∩ U^{g} ≠ ∅ ⇒ (fg)(u) = g(v) for some u, v ∈ U ⇒ v ∈ U^{e} ∩ U with e = g^{−1}fg ∈ G ⇒ (because U is a block) e(U) = U ⇒ (by applying g to both sides) (fg)(U) = g(U) ⇒ (U^{g})^{f} = U^{g}. Therefore U^{g} is a block, and similarly so is U^{h}. Moreover: U^{g} ∩ U^{h} ≠ ∅ ⇒ g(u) = h(v) for some u, v ∈ W ⇒ v ∈ U^{e} ∩ U with e = h^{−1}g ∈ G ⇒ (because U is a block) e(U) = U ⇒ (by applying h to both sides) g(U) = h(U) ⇒ U^{g} = U^{h}.

(5.7.4) is obvious, and (5.7.5) follows from (5.7.3). Moreover, (5.7.6) follows from (5.7.4) and (5.7.5). Finally, to prove (5.7.7) assume that G is transitive and let N ◁ G. Then for any g ∈ G we have N^{g} = N and hence by taking N for H in (5.7.1) we get (u^{N})^{g} = (u^{g})^{N} for all u ∈ W. Therefore by taking u^{N} for U in (5.7.5) we see that the set of all N-orbits is a G-invariant partition, and hence N is semi-transitive and the size of any N-orbit divides n.

With these preliminaries at hand, we proceed to prove the following lemma, where in connection with (5.7.1) we note that a maximal subgroup of a group is a subgroup which is not contained in any subgroup other than itself or the whole group; in connection with (5.7.1) also note that, in view of the Conjugacy Lemma, the phrase “for every u ∈ W ” may be replaced by the phrase “for some u ∈ W ”.

Lemma 5.8 (Primitivity Lemma) We have the following.

5.8.1 G primitive ⇔ G transitive and has no nontrivial block ⇔ G transitive and G_{u} maximal in G for every u ∈ W.

5.8.2 G two-transitive ⇒ G primitive.

5.8.3 G sesqui-transitive and imprimitive ⇒ G semi-Frobenius.

The first implication in (5.8.1) follows from (5.7.5) and (5.7.6). To prove the second implication, first assume that G is transitive but has no nontrivial block, and let G_{u} < H < G with u ∈ W; then by (5.7.2) we see that u^{H} is a G-block with |u^{H}| = [H : G_{u}], and hence [H : G_{u}] = 1 or n; the transitivity of G tells us that [G : G_{u}] = n, and therefore we must have H = G_{u} or G. Conversely, suppose that G is transitive and has a nontrivial block U; then we can find u ≠ v in U, and w in W \ U; now upon letting H = stab_{G}(U) = {g ∈ G : U^{g} = U} we have H < G, and by the blockness of U we get G_{u} < H; since u ≠ v in U, by the transitivity of G we find h ∈ G with h(u) = v and then h ∉ G_{u} but by the blockness of U we get U^{h} = U and hence h ∈ H and therefore G_{u} ≠ H; since u ≠ w in W with u ∈ U and w ∉ U, by the transitivity of G we find f ∈ G with f(u) = w and then f ∉ H and hence H ≠ G; thus G_{u} is not maximal in G.

To prove (5.8.2) assume that G is two-transitive. Let if possible, U be a nontrivial G-block. Then first by the nontriviality we can find u ≠ v in U and w ∈ W \ U, and then by the two-transitivity we can find g ∈ G such that g(u) = u and g(v) = w. Now U^{g} ∩ U ≠ ∅ because u ∈ U^{g} ∩ U, and U^{g} ≠ U because w ∈ U^{g} \ U. This contradicts the blockness of U. Therefore G is primitive by (5.8.1).

Finally, to prove (5.8.3) assume that G is sesqui-transitive and imprimitive. Then there is a G-invariant partition T = (T_{i})_{1≤i≤r} of W of type (r, s) with r > 1 and s > 1 and T_{i} = {w_{i1}, …, w_{is}} where w_{i1}, …, w_{is} are distinct points of W. Given any u ≠ v in W, upon letting H = G_{u}, we want to show that H_{v} = 1. By relabelling the T_{i} and the w_{ij} we may assume that u = w_{11}. Since G is sesqui-transitive, all the H-orbits on W \ {u} have the same size, say m. Since W \ {u} is a disjoint union of the H-orbits on it, m divides |W \ {u}| = rs − 1. Therefore GCD(m, s) = 1. Fix any i with 1 < i ≤ r. Then clearly u ∉ (T_{i})^{H} and ((T_{i})^{H})^{h} = (T_{i})^{H} for all h ∈ H, and hence (T_{i})^{H} is a disjoint union of some H-orbits on W \ {u} and therefore m divides |(T_{i})^{H}|. Since T is a G-invariant partition, (T_{i})^{H} is also the disjoint union of some of the T_{j}, and hence |(T_{i})^{H}| is divisible by s. Since GCD(m, s) = 1, it follows that ms divides |(T_{i})^{H}|. Now we have the union (T_{i})^{H} = (w_{i1})^{H} ∪ ⋯ ∪ (w_{is})^{H} with |(w_{i1})^{H}| = ⋯ = |(w_{is})^{H}| = m and hence |(T_{i})^{H}| ≤ ms; since ms divides |(T_{i})^{H}|, we must have |(T_{i})^{H}| = ms and the said union must be disjoint. Since T_{i} = {w_{i1}, …, w_{is}} and the H-orbits (w_{i1})^{H}, …, (w_{is})^{H} are pairwise disjoint, we must have: (*) (w_{ij})^{H} ∩ T_{i} = {w_{ij}} for 1 ≤ j ≤ s. If v ∈ T_{i} and h ∈ H_{v}, then the G-blockness of T_{i} tells us that (T_{i})^{h} = T_{i} and, for 1 ≤ j ≤ s, we clearly have ((w_{ij})^{H})^{h} = (w_{ij})^{H}, and hence by (*) we get (w_{ij})^{h} = ((w_{ij})^{H} ∩ T_{i})^{h} = ((w_{ij})^{H})^{h} ∩ (T_{i})^{h} = (w_{ij})^{H} ∩ T_{i} = {w_{ij}}; since i was any integer with 1 < i ≤ r and h was any element of H_{v}, we conclude that: (**) v ∈ T_{i} with i ∈ {2, …, r} ⇒ H_{v} < G_{T_{i}} where we recall that G_{T_{i}} = {g ∈ G : t^{g} = t for all t ∈ T_{i}}. Clearly H_{v} = (G_{v})_{u}, and hence by interchanging u and v in (**) we get: (1) v ∈ T_{i} with i ∈ {2, …, r} ⇒ H_{v} < G_{T_{1}}. If v ∈ T_{i} then, for 2 ≤ j ≤ s, obviously G_{T_{1}} < H_{w_{1j}} and hence by (1) we get H_{v} < H_{w_{1j}}; again since i was any integer with 1 < i ≤ r, we conclude that: (2) v ∈ T_{i} with i ∈ {2, …, r} ⇒ H_{v} = H_{w_{12}} = ⋯ = H_{w_{1s}}. The only property of v used in (2) was that v ∉ T_{1}; consequently: (3) H_{w_{12}} = ⋯ = H_{w_{1s}} = H_{w} for all w ∈ W \ T_{1}. By (3) we see that: (1′) H_{w_{1j}} = 1 for 2 ≤ j ≤ s. By (2) and (1′) we see that: (2′) v ∈ T_{i} with i ∈ {2, …, r} ⇒ H_{v} = 1. Finally by (1′) and (2′) we conclude that we always have H_{v} = 1.

Remark 5.9 Before proceeding further, let us observe some general things about a group X. Recall that a minimal normal subgroup of X is a minimal element in the set of all nonidentity normal subgroups of X; if X is a nonidentity finite group then clearly X has minimal normal subgroups. Also recall that a characteristic subgroup of X is a subgroup which is mapped onto itself by every automorphism of X. All “group theoretically well-defined” subgroups are characteristic subgroups; for instance:

5.9.1 the center of X is a characteristic subgroup of X; similarly:

5.9.2 if X is finite and has a unique p-Sylow subgroup for some prime divisor p of |X| then it is also a characteristic subgroup of X; more generally:

5.9.3 if X is finite and p is any prime then the subgroup of G generated by all of its p-Sylow subgroups (which is denoted by p(G) and which may be defined as the subgroup of G generated by all of its elements of p-power order) is a characteristic subgroup of X; likewise:

5.9.4 if X has a unique minimal normal subgroup then it is a characteristic subgroup of X; more generally:

5.9.5 the subgroup of X generated by all of its minimal normal subgroups is a characteristic subgroup of X. Moreover:

5.9.6 if Y is a minimal normal subgroup of X, and Z is a nonidentity characteristic subgroup of Y, then (clearly Z is normal in X and hence by minimality) we must have Y = Z. Finally:

5.9.7 if Y is a minimal normal subgroup of a finite group X, and Y has a unique minimal normal subgroup Z, then (by (5.9.4) and (5.9.6) we get Y = Z and by minimality Z is contained in every nonidentity normal subgroup of Y and hence) Y must be a simple group.

Next we shall prove the:

Lemma 5.10 (Normality Lemma) For any 1 ≠ N◁ G we have the following:

5.10.1 G primitive ⇒ N transitive.

5.10.2 G two-transitive ⇒ N sesqui-transitive.

5.10.3 G two-transitive and N regular ⇒ N elementary abelian.

To prove (5.10.1), note that if N ◁ G with G primitive then, by (5.7.7), either all the N-orbits are singletons or W is the only N-orbit; if N ≠ 1 then the first alternative is not possible and hence W is the only N-orbit, i.e., N is transitive.

To prove (5.10.2) assume that 1 ≠ N ◁ G with G two-transitive. Then N is transitive by (5.8.2) and (5.10.1). For any u ∈ W we have N_{u} = N ∩ G_{u} ◁ G_{u} with G_{u} transitive on W \ {u}, and hence by (5.7.7) we see that N_{u} is semi-transitive on W \ {u}. Therefore N is sesqui-transitive.

Finally, to prove (5.10.3), let N ◁ G be such that N is regular. Fixing any w ∈ W, let f : W → N be the bijection whose inverse is given by g ↦ g(w), and let f^{♯} : Sym(W) → Sym(N) be the isomorphism induced by f. Then, for any h ∈ G_{w} and g ∈ N, upon letting u = g(w) ∈ W and v = h(u) ∈ W and g^{∗} = f(v) ∈ N, by the definition of f we have f(u) = g and g^{∗}(w) = v, and hence by the definition of f^{♯} we get f^{♯}(h)(g) ∈ N with (f^{♯}(h)(g))(w) = v, and by taking (g, h, w) for (h, g, u) in the Conjugation Rule (5.4) we also get g^{h} ∈ N with g^{h}(w) = v; since g ↦ g(w) gives a bijection N → W, we must have f^{♯}(h)(g) = g^{h}. Thus, for every h ∈ G_{w}, the permutation f^{♯}(h) : N → N coincides with the conjugation automorphism given by g ↦ g^{h} and hence the order of any g ∈ N equals the order of f^{♯}(h)(g) ∈ N. For any g ≠ 1 ≠ g′ in N we have g(w) ≠ w ≠ g′(w) and, assuming G to be two-transitive, we can find h ∈ G_{w} with h(g(w)) = g′(w), and then clearly f^{♯}(h)(g) = g′, and hence the order of g equals the order of g′. Let p > 1 be the common order of all elements of N \ {1}. If p = rs with r > 1 and s > 1 then taking 1 ≠ g ∈ N we get 1 ≠ g^{r} ∈ N and the order of g^{r} is s with 1 < s < p which is a contradiction. Therefore p must be prime. If the order of N were not a power of p then by Sylow’s theorem N would contain an element whose order is a prime different from p which would be a contradiction. Therefore the order of N must be a power of p, and hence N has a nontrivial center M; obviously M is a characteristic subgroup of N and hence an automorphism of N cannot send an element in M to one not in M; but we have just shown that any two nonidentity elements of N are conjugate in G; therefore we must have M = N, i.e., N is abelian. Since the order of every nonidentity element of N is p, it follows that N is elementary abelian.

As a consequence of the Orbit-Stabilizer Equation (5.2), let us prove the following Lemma, where we note that |X|_{p} denotes the highest power of a prime p which divides the size |X| of a nonempty finite set X and, in case X is a finite group, by a p-Sylow subgroup of X we mean a subgroup of X whose order equals |X|_{p}.

Lemma 5.11 (Sylow Transitivity Lemma) Let X be a finite group acting on the finite set W, i.e., we are given a homomorphism θ : X → Sym(W). For every u ∈ W let stab_{X}(u) = {x ∈ X : θ(x) ∈ stab_{θ(X)}(u)} and orb_{X}(u) = orb_{θ(X)}(u). Then we have the following, where in (5.11.1) to (5.11.4) u is any given point of W, in (5.11.2) to (5.11.5) p is any given prime, in (5.11.2) to (5.11.4) |Z|_{p} denotes the highest power of p which divides the size |Z| of a finite set Z, and in (5.11.5) by a p-power X-orbit in W we mean a subset of W whose size is a power of p and which is of the form orb_{X}(w) for some w ∈ W.

5.11.1 |X|/|stab_{X}(u)| = |orb_{X}(u)|.
5.11.2 |X|_{p}/|stab_{X}(u)|_{p} = |orb_{X}(u)|_{p}.

5.11.3 Let Y < X be such that |Y |_{p} = |X|_{p} (for instance Y could be a p-Sylow subgroup of X). Then |orb_{Y} (u)|_{p} ≥ |orb_{X}(u)|_{p}.

5.11.4 In the situation of (5.11.3) assume that |orb_{X}(u)| = a power of p. Then Y acts transitively on orb_{X}(u), i.e., orb_{X}(u) = orb_{Y} (u).

5.11.5 Every p-Sylow subgroup of X acts transitively on every p-power X-orbit in W.

Namely, (5.11.1) follows by taking G = θ(X) in (5.2) and noting that then obviously orb_{X}(u) = u^{G} and also ker θ < stab_{X}(u) and hence |X|/|stab_{X}(u)| = |θ(X)|/|θ(stab_{X}(u))| = |G|/|G_{u}|. To prove (5.11.2) note that for any Z < X we clearly have |X|_{p}/|Z|_{p} = the highest power of p which divides |X|/|Z|, and now apply (5.11.1) with Z = stab_{X}(u). To prove (5.11.3) note that obviously stab_{Y} (u) = Y ∩ stab_{X}(u), and hence |stab_{Y} (u)| divides |stab_{X}(u)|, and therefore |stab_{Y} (u)|_{p} divides |stab_{X}(u)|_{p}, and hence our assertion follows from (5.11.2). To prove (5.11.4) note that

|orb_{Y} (u)| ≥ |orb_{Y} (u)|_{p} ≥ |orb_{X}(u)|_{p} = |orb_{X}(u)|

where the first inequality is obvious, the second inequality is (5.11.3), and the third equation is equivalent to the assumption of |orb_{X}(u)| being a power of p; therefore |orb_{Y} (u)| ≥ |orb_{X}(u)|; but obviously |orb_{Y} (u)| ≤ |orb_{X}(u)|; therefore |orb_{Y} (u)| = |orb_{X}(u)| and hence orb_{Y} (u) = orb_{X}(u). Finally (5.11.5) follows from (5.11.4) by taking Y to be the given p-Sylow subgroup of X, and orb_{X}(u) to be the given p-power X-orbit in W.

Now we shall prove the:

Lemma 5.12 (Fixed Point Lemma) Assume that G is semi-Frobenius, and let N = {1} ∪ N′ where N′ is the set of all fixed point free elements of G. Then we have the following:

5.12.1 |N| = n, and for each prime divisor p of n we have that N has an element of order p, and every element of G whose order is a power of p belongs to N.

5.12.2 If all elements of N′ have equal order, then that order is a prime p, n is a power of p, N is the unique p-Sylow subgroup of G, N is a characteristic subgroup of G, and N is regular.

5.12.3 If G ◁ G^{∗} < Sym(W) with G^{∗} two-transitive, then all elements of N′ have equal order which is a prime p, n is a power of p, N is the unique p-Sylow subgroup of G, N is a characteristic subgroup of G, N is regular, and N is elementary abelian.

5.12.4 If G ◁ G^{∗} < Sym(W) with G^{∗} two-transitive and G minimal normal in G^{∗}, then G is regular and G = N = an elementary abelian group.

To prove this, first note that, assuming G to be semi-Frobenius, and letting w_{1}, …, w_{n} be the distinct elements of W, by (5.3.4) and (5.3.6) we have |G| = mn where m is a divisor of n − 1 with |G_{w_{i}}| = m for 1 ≤ i ≤ n. Since G is semi-Frobenius, we have G_{w_{i}} ∩ G_{w_{j}} = {1} for all i ≠ j, and hence |∪_{1≤i≤n}G_{w_{i}}| = 1 + (m − 1)n. Upon letting N = {1} ∪ N′ where N′ is the set of all fixed point free elements of G, we clearly have N′ = G \ ∪_{1≤i≤n}G_{w_{i}}, and hence |N′| = mn − [1 + (m − 1)n], and therefore |N| = 1 + |N′| = n. Since the order of G_{w_{i}} divides n − 1, the order of any nonidentity element of G_{w_{i}} must divide n − 1 and hence it cannot divide n; since N′ = G \ ∪_{1≤i≤n}G_{w_{i}}, every nonidentity element of G whose order divides n must belong to N′; therefore N contains every element of G whose order divides n. Since n divides |G|, by Sylow’s Theorem G does have elements of every prime order dividing n. Consequently, for each prime divisor p of n we have that N has an element of order p, and every element of G whose order is a power of p belongs to N. This completes the proof of (5.12.1).

To prove (5.12.2) assume that all elements of N′ have the same order, say p. Then by (5.12.1), p must be prime and n must be a power of p. Since |G| = mn with GCD(m, n) = 1, n must be the highest power of p dividing |G|. By (5.12.1), |N| = n and N contains every element of G whose order is a power of p. Therefore N must be the unique p-Sylow subgroup of G. Consequently N is a characteristic subgroup of G. Since G transitive, W is an orbit of G; since |W | = a power of p, and N is a p-Sylow subgroup of G, by the Sylow Transitivity Lemma (5.11.5) we see that N is transitive on W; therefore, since |N| = n, by (5.3.5) we conclude that N is regular.

To prove (5.12.3) assume that G ◁ G^{∗} < Sym(W) with G^{∗} two-transitive. Let A = {(h, u, v) ∈ N′ × W × W : h(u) = v}. Then (h, u, v) ↦ (h, u) gives a bijection A → N′ × W, and |W | = n and by (5.12.1) we have |N′| = n − 1; therefore |A| = (n − 1)n. Let us fix some (h, u, v) ∈ A. Upon letting B = {(u′, v′) ∈ W × W : u′ ≠ v′}, given any (u′, v′) ∈ B, by the two-transitivity of G^{∗} we can find g ∈ G^{∗} with g(u) = u′ and g(v) = v′, and now by the Conjugation Rule (5.4) and the normality of G we get (h^{g}, u′, v′) ∈ A; for each (u′, v′) ∈ B we fix such g ∈ G^{∗} and then (u′, v′) ↦ (h^{g}, u′, v′) gives an injection B → A; but |B| = (n − 1)n = |A| and hence the said injection must be a bijection. Therefore {h^{g} : g ∈ G^{∗}} = N′. Since conjugate elements have the same order, it follows that all elements of N′ have equal order, say p. Now by (5.12.2) it follows that p is prime, n is a power of p, N is the unique p-Sylow subgroup of G, N is a characteristic subgroup of G, and N is regular. Since N is a regular normal subgroup of the two-transitive group G^{∗}, by (5.10.3) we see that N is elementary abelian.

Finally, if G ◁ G^{∗} < Sym(W) with G^{∗} two-transitive and G minimal normal in G^{∗}, then by (5.12.3) we see that N is regular, elementary abelian, and a characteristic subgroup of G, and hence by (5.9.6) we get G = N. This completes the proof of (5.12.4).

Next we shall prove the:

Lemma 5.13 (Double Normality Lemma) Assume that G has two distinct minimal normal subgroups N and N′. Then we have the following:

5.13.1 N ∩ N′ = 1, N and N′ centralize each other, and NN′ = N × N′ ◁ G where the product is direct.

5.13.2 If G is primitive, then N and N′ are the centralizers of each other in Sym(W), N and N′ are regular, N and N′ are nonabelian groups, G has no minimal normal subgroup other than N and N′, and NN′ is a characteristic subgroup of G with |NN′| = n^{2}. [Now by (5.6.3) it follows that N and N′ are permutationisomorphic and, in the sense explained there, they are the left and right regular representations of each other].

5.13.3 G cannot be two-transitive.

5.13.4 If G◁G^{∗} < Sym(W) with G primitive and minimal normal in G^{∗}, then G^{∗} cannot be two-transitive.

To prove this assume that G has two distinct minimal normal subgroups N and N′. Then obviously N ∩ N′ ◁ G with and hence by the minimality of N and N′ we get N ∩ N′ = 1, and therefore NN′ = N × N′ ◁ G where the product is direct. Now the commutator group [N, N′] is the subgroup of G generated by all the commutators ghg^{−1}h^{−1} with g ∈ N and h ∈ N′; clearly gh = hg ⇔ ghg^{−1}h^{−1} = 1, and hence: [N, N′] = 1 ⇔ N and N′ centralize each other. For all g ∈ N and h ∈ N′ by the normality of N′ we have ghg^{−1} ∈ N′ and hence ghg^{−1}h^{−1} ∈ N′, and similarly by the normality of N we get ghg^{−1}h^{−1} ∈ N; therefore [N, N′] < N ∩ N′; consequently [N, N′] = 1 and hence N and N′ centralize each other. This proves (5.13.1).

To prove (5.13.2) assume that G is primitive. Then by (5.10.1) we know that N and N′ are transitive, and hence by (5.6.2) we see that N and N′ are the centralizers of each other in Sym(W), N and N′ are regular, and N′ is the only transitive subgroup of Sym(W) which centralizes N. Since N ∩ N′ = 1 and N and N′ are the centralizers of each other, they must be nonabelian. As we have just seen, by (5.10.1) and (5.6.2) it follows that if N′′ is any minimal normal subgroup of G with N′′ ≠ N such that N′′ centralizes N then N′′ is the only transitive subgroup of Sym(G) which centralizes N, and since (as said above) N′ is also the only transitive subgroup of Sym(G) which centralizes N, we must have N′′ = N. Thus G has no minimal normal subgroups other than N and N′, and hence NN′ is a characteristic subgroup of G. Since N and N′ are regular, by (5.3.5) we get |N| = |N′| = n; since NN′ is the direct product of N and N′, we must have |NN′| = n^{2}. This completes the proof of (5.13.2), and the bracketed remark at the end of it follows from (5.6.3).

If G were two-transitive, then by (5.8.2) G would be primitive, and hence by (5.13.2) N would be a nonabelian regular normal subgroup of G which would contradict (5.10.3). This proves (5.13.3).

To prove (5.13.4) let G ◁ G^{∗} < Sym(W) with G primitive and minimal normal in G^{∗}. Then by (5.13.2) we know that NN′ is a characteristic subgroup of G, and hence by (5.9.2) we get NN′ = G. By (5.13.2) we have |NN′| = n^{2}, and hence by (5.3.7) we see that G cannot be sesqui-transitive, and therefore by (5.10.2) we conclude that G^{∗} cannot be two-transitive.

Now we shall prove:

Lemma 5.14 (Burnside’s Lemma) Assuming G to be two-transitive, for any minimal normal subgroup N of G we have the following.

5.14.1 N is primitive or semi-Frobenius.

5.14.2 If N is semi-Frobenius then it is regular.

5.14.3 If N is regular then it is elementary abelian.

5.14.4 If N is primitive abelian simple then it is regular.

5.14.5 If N is primitive then it is simple.

To prove this assume that G is two-transitive and let N be a minimal normal subgroup of G. Then by (5.10.2) N is sesqui-transitive, and hence by (5.8.3) it is primitive or semi-Frobenius, which proves (5.14.1). (5.14.2) follows by taking (N, G) for (G, G^{∗}) in (5.12.4). (5.14.3) follows from (5.10.3). (5.14.4) follows by noting that if N is primitive then N is transitive and by (5.3.4) and (5.8.1) we see that for any u ∈ W we have N_{u} ≠ N (because n > 1 by two-transitivity of G) and there is no group H with N_{u} < H < N and N_{u} ≠ H ≠ N, and hence if N is also abelian simple (i.e., of prime order) then we must have N_{u} = 1, i.e., N is regular. Finally, (5.14.5) follows by noting that if N is primitive then by taking (N, G) for (G, G^{∗}) in (5.13.4) we see that N has a unique minimal normal subgroup, and hence by (5.9.7) we conclude that N is simple.

Finally we are ready to prove:

Theorem 5.15 (Burnside’s Theorem) A two-transitive permutation group G has a unique minimal normal subgroup N. Moreover, if N is regular then it is elementary abelian, and if N is nonregular then it is primitive as well as nonabelian simple.

Namely, if G is two-transitive then by (5.13.3) we see that G has a unique minimal normal subgroup N. The rest follows from (5.14).

Remark 5.16 Let N = {1} ∪ N′ where N′ is the set of all fixed point free elements of G. Recall that G is semi-Frobenius means G is transitive and has no nonidentity element fixing two points, and G is Frobenius means G is semi-Frobenius but not regular. In the Fixed Point Lemma (5.12.4) we have proved that if G is semi-Frobenius and G◁G^{∗} < Sym(W) with G^{∗} two-transitive and G minimal normal in G^{∗}, then G is regular and G = N = an elementary abelian group. This motivates Frobenius’ Theorem, proved in his 1901 paper [23], which says that if G is Frobenius then N is a regular normal subgroup of G. It is clear that Frobenius’ Theorem implies the Fixed Point Lemma (5.12.4), and this gives rise to the proof of Burnside’s Theorem using Frobenius’ Theorem.

6 Analytic Geometry

In analytic geometry, we put equations into good shape by choosing a good coordinate system. Thus an equation of a circle, when we bring its center to the origin and do suitable rescaling, becomes x^{2} + y^{2} = 1. Similarly a sphere becomes x^{2} +y^{2} +z^{2} = 1. Likewise, by translation and rotation, an ellipse becomes \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1. In the plane, to change coordinates, we use a transformation of the form

x ↦ ax + by + c
y ↦ αx + βy + γ

where ↦ is to be read as goes to, or maps to. We usually don’t pay much attention to the one-dimensional version of this transformation, i.e.,

x ↦ ax + b.

Yet, this transformation is more important than the plane transformation. One reason for the importance of x ↦ ax + b is that in it we may interpret a as an m × m matrix, and x and b as column vectors. When we do that, we have an affine linear transformation of m-space. Moreover, as we shall see in a moment, hidden in x ↦ ax + b are the ideas of semidirect product and the regular representation of Cayley’s theorem.

Now where do the values of a, x, and b come from? Well, they may be rational numbers, or real numbers, or complex numbers, which are all examples of a field. Quite generally, a field k is a set of elements with the operations of addition and multiplications defined on them. Under addition, k is a commutative group which we call the additive group of k and denote it by k^{+}. Under multiplication, the nonzero elements of k form a group which we call the multiplicative group of nonzero elements of k and denote it by k^{×}. At any rate, the values of a, x, and b can come from any field k with the understanding that a is nonzero. By taking a = 1, we get the translations x ↦ x + b which constitute the regular representation N of the additive group k^{+} of the underlying field k, which is what Cayley generalized into the regular permutation representation of an arbitrary group which need not be commutative. Likewise, taking b = 0, we get the group A of all transformations x ↦ ax with a varying in the multiplicative group k^{×} of all nonzero elements of k. Thus N ≈ k^{+} and A ≈ k^{×} where ≈ denotes isomorphism. Now every element of the group H of all transformations x ↦ ax + b can be uniquely expressed as a product of an element of N and an element of A. Thus H is the prototype of the semidirect product of a group N by a group A. Thinking of A as a subgroup of the group Aut(N) of all automorphisms of N, the group H may be regarded as the partial holomorph of N relative to A. Quite generally, the holomorph of any group N is formed by putting together N and Aut(N) as subgroups of a common group. Now the entire group H, as a permutation group on k, is doubly transitive, i.e., it sends any two elements of k to any two elements of k. Burnside’s Theorem says that in the finite case the converse is true, i.e., any doubly transitive finite permutation group has a unique minimal normal subgroup, which is either elementary abelian or simple; thus in the example of H, the normal subgroup N is abelian, and so it only corresponds to half of Burnside’s Theorem. Burnside’s Theorem is the foundation of CDT; CDT is the flowering of Burnside’s Theorem. The half of CDT which corresponds to the above half of Burnside’s Theorem is CPT. At any rate, the secret to understanding CDT is to understand x ↦ ax + b.

To generalize the above discussion, let us replace N by an arbitrary, possibly noncommutative, group X which as usual is written multiplicatively. Let there be given any A < Aut(X), i.e., a subgroup A of the group Aut(X) of all automorphisms of X. Note that Aut(X) < Sym(X) = (the symmetric group on X) = (the group of all bijections of X). For any a ∈ A and b ∈ X, let Φ_{a,b} be the bijection of X under which the value of any x ∈ X is obtained by multiplying a(x) from the left by b, i.e., Φ_{a,b}(x) = ba(x); note that if X is an additive abelian group then this reduces to the more familiar form Φ_{a,b}(x) = a(x) + b. Now we define the (relative) holomorph H(X, A) of (X, A) to be the subgroup of Sym(X) consisting of all Φ_{a,b} with a varying in A and b varying in X. Note that b ↦ Φ_{1,b} gives the left multiplication map of X into Sym(X), which we may call the left Cayley map of X, and for its image H(X, 1) we have H(X, 1) ◁ H(X, A), i.e., H(X, 1) is a normal subgroup of H(X, A). Moreover H(X, A) is the semidirect product of H(X, 1) and A, where we recall that a group H′ is the semidirect product of X′ and A′ means X′ ◁ H′ and A′ < H′ with X′ ∩ A′ = 1, and H′ is generated by the subgroups X′ and A′. Finally we define the (full) holomorph H(X) of X by putting H(X) = H(X, Aut(X)).

Now to formalize the definition of affine linear transformations, we recall that the group of all nonsingular linear transformations of a finite dimensional vector space V over a field k is called the general linear group of V and is denoted by GL(V), and we define the affine general linear group AGL(V) of V by putting AGL(V) = H(V, GL(V)).

7 Cayley Revisited

A permutation on n letters u_{1}, …, u_{n} which rearranges them as v_{1}, …, v_{n} is usually indicated by the two-rowed matrix

(\begin{matrix}u_{1} & … & u_{i} & … & u_{n} \\ v_{1} & … & v_{i} & … & v_{n} \end{matrix})

which says that u_{i} is mapped to v_{i}. These n! permutations constitute the symmetric group S_{n} = Sym({u_{1}, …, u_{n}}).

For any finite group X = {x_{1}, …, x_{n}} of order n, Cayley’s Theorem provides an isomorphic copy of X inside S_{n} = Sym(X). Indeed it provides two of them, one by the left multiplication L_{X} : X → Sym(X) which sends any z ∈ X to

L_{X}(z) = (\begin{matrix}x_{1} & … & x_{i} & … & x_{n} \\ zx_{1} & … & zx_{i} & … & zx_{n} \end{matrix})

and the other by the right multiplication R_{X} : X → Sym(X) which sends any y ∈ X to

R_{X}(y) = (\begin{matrix}x_{1} & … & x_{i} & … & x_{n} \\ x_{1}y^{−1} & … & x_{i}y^{−1} & … & x_{n}y^{−1} \end{matrix})

where we multiply by y^{−1} to make up for the fact that we are composing maps from right to left, i.e., we are writing (ϕψ)(x) = ϕ(ψ(x)). These two copies of X, i.e., L_{X}(X) and R_{X}(X), although equivalent (= permutation-isomorphic), are not the same.

Many group theory books discuss only one of these, perhaps because they compose either from left to right or right to left. So they miss out on the facts that L_{X}(X) and R_{X}(X) centralize each other, and indeed are the full centralizers of each other in S_{n}. Out of this the first fact is obvious because left multiplication by z and right multiplication by y^{−1} do not interfere with each other and hence, as maps, they commute with each other. To prove the second fact, let η ∈ Sym(X) be such that η commutes with every member of L_{X}(X); then for all z and x in X we have η(zx) = zη(x), and by putting x = 1 in this equation we get η(z) = zy^{−1} where y = η(1)^{−1}, and hence η = R_{X}(y); similarly any ζ ∈ Sym(X) which commutes with every member of R_{X}(X) must belong to L_{X}(X).

Thus the left and right multiplications give us the two regular representations L_{X}(X) and R_{X}(X) of X which centralize each other. Heuristically we can say that since there is no third side to multiply from, nothing else can centralize them. In the Centralizer Lemma (5.6) we have made this heuristic more precise, and from it, in the second part of the Double Normality (5.13) we have deduced the fact that a primitive permutation group cannot have more than two minimal normal subgroups, and if it did have two then they must be nonabelian and the left and right regular representations of each other. Recall that primitivity is something between transitivity and two-transitivity, and note that from the second of (5.13) we deduced its third part which says that a two-transitive permutation group G has only one minimal normal subgroup N. In this deduction we used the Normality Lemma (5.10.3) which says that a regular normal subgroup of a two-transitive permutation group must be elementary abelian. Applying this same principle that sometimes a permutation group cannot accommodate two minimal normal subgroups inside its stomach to N itself, in Burnside’s Lemma (5.14) we showed that if N is not regular then it must be simple. Thus, in effect, the entire proof of Burnside’s Theorem is covered by the multifaceted ramifications of Cayley’s Theorem.

Remark 7.1 Recall that a permutation group N is regular means, for every pair of points u, v in the permuted set, exactly one member of N sends u to v. In the Centralizer Lemma (5.6) we dealt with any regular permutation group N < S_{n} = Sym(W) where W is a nonempty finite set of size n, and fixing any w ∈ W, we considered the bijection f : W → N whose inverse is given by g ↦ g(w). This induced the isomorphism f^{♯} : Sym(W) → Sym(N) given by f^{♯}(h)(f(u)) = f(h(u)) for all h ∈ Sym(W) and u ∈ W. We noted that then f^{♯}(N) = L_{N}(N) and f^{♯}(N′) = R_{N}(N) where N′ is the centralizer of N in Sym(W).

8 Burnside Revisited

The Normality Lemma (5.10.3) says that a regular normal subgroup of a twotransitive permutation group must be elementary abelian. The proof of this given in Section 5, although it too came out of a further unravelling of Cayley’s Theorem, was perhaps a bit too condensed. So let us now expand on it by proving the following Results.

Result 8.1 Let X be a finite group. Assume that X ≠ 1, and there exists A < Aut(X) such that A acts transitively on X \ {1}. Then X is elementary abelian.

Proof To see this, we note that since an automorphism must map an element of some order to an element of the same order, and since A acts transitively on X \ {1}, all elements of X \ {1} must have the same order. If this order were not prime, then we could take a nonidentity element of X and raise it to a power, the exponent being a nontrivial factor of the order, so that the new element so obtained would have an order less than the common order. So the common order must be a prime p. Now by Cauchy’s Theorem (= pre-Sylow Theorem) we see that |X| must be a power of p. Therefore by a well-known elementary result (which, as in [12, 5.14], follows from our (5.2)), the center of a nonidentity p-power order group is nonidentity. Therefore for the center Y of X we have Y ≠ 1. But the center of any group is obviously a characteristic subgroup, where we recall that a subgroup is called characteristic if it is mapped onto itself by every automorphism of the group. Therefore Y is a characteristic subgroup of X. If Y ≠ X, then we can take 1 ≠ y ∈ Y and z ∈ X \ Y; since A acts transitively on X \ {1}, we can find a ∈ A with a(y) = z; but since Y is a characteristic subgroup of X, we must have a(Y) = Y which is a contradiction. Therefore Y = X; since Y is the center of X, it follows that X is abelian; but since every nonidentity element of it has order p, it must be elementary abelian.

Result 8.2 Let N ◁ G < S_{n} = Sym(W) where W is a finite set of size n > 1, and N is regular. Fixing any w ∈ W let G_{w} be the stabilizer of w in G, i.e., G_{w} = {h ∈ G : h(w) = w}. Then G is the semidirect product of N and G_{w}, i.e., N ∩ G_{w} = 1 and G is generated by N and G_{w}. Moreover, in the notation of (7.1), for every h ∈ G_{w} the permutation f^{♯}(h) : N → N coincides with the conjugation automorphism g ↦ g^{h}, i.e., for all h ∈ G_{w} and g ∈ N we have f^{♯}(h)(g) = g^{h} where g^{h} = hgh^{−1}.

Proof Clearly N ∩ G_{w} = 1, and for any h ∈ G upon letting g to be the unique member of N with g(w) = h(w) we get (g^{−1}h)(w) = w and hence h = gh′ with h′ = g^{−1}h ∈ G_{w}, which shows that G is generated by N and G_{w}. The second assertion starting with “Moreover” follows as in the first half of the proof of (5.10.3).

Remark 8.3 In the situation of (8.2), if G is two-transitive then as in the middle part of the proof of (5.10.3) we see that the conjugation action of G_{w} on N is transitive on N \ {1}, and hence by (8.1) we conclude that N is elementary abelian, which reproves (5.10.3).

Remark 8.4 In the situation of (8.2) assume that N is elementary abelian; by (8.3) we know this to be so in case G is two-transitive. Note that N being elementary abelian is equivalent to saying that, for some positive integer m and some prime p, we have an isomorphism f′ of the multiplicative group N onto the additive group of an m dimensional vector space V over the prime field GF(p); also note that GL(V) = Aut(V) < Sym(V). Now we have the bijection f^{∗} = f′f : W → V, which induces the isomorphism f^{∗♯} : Sym(W) → Sym(V) given by f^{∗♯}(h)(f^{∗}(u)) = f^{∗}(h(u)) for all h ∈ Sym(W) and u ∈ W. Upon letting A = f^{∗♯}(G_{w}), and continuing the discussion started in the Section on Analytic Geometry, we see that f^{∗♯}(G) = H(V, A) < AGL(V) with A < GL(V) = Aut(V) and f^{∗♯}(N) = H(V, 1) ◁ H(A, V). Thus G is permutation-isomorphic to a subgroup of AGL(V) = H(V, GL(V)) of the form H(V, A) with A < GL(V) where, if G is twotransitive then A is transitive on nonzero vectors, i.e., on V \{0}. Conversely, if A < GL(V) is transitive on nonzero vectors then H(V, A) is a two-transitive subgroup of AGL(V), the stabilizer of the zero vector in H(A, V) is A, and H(V, 1) is a regular normal subgroup of H(V, A). Thus the category of two-transitive permutation groups having a regular normal subgroup is equivalent to the category of groups H(V, A) with finite positive dimensional vector spaces V over finite prime fields together with subgroups A of GL(V) acting transitively on nonzero vectors. We could also say that, in the case of an elementary abelian group V, Cayley’s regular group H(V, 1) = L_{V} (V) = R_{V} (V), when bedecked with the partial halo of A < Aut(V) acting transitively on nonzero vectors, becomes the group H(V, A) which is Burnside’s two-transitive group with a regular normal subgroup. Likewise, in the general case of a not necessarily abelian group X, Cayley’s regular group H(X, 1) = L_{X}(X), when bedecked with the fuller halo of Aut(X), becomes the holomorph H(X, Aut(X)).

Remark 8.5 Recall that a group H′ is the semidirect product of X′ and A′ means X′ ◁H′ and A′ < H′ with X′ ∩A′ = 1, and H′ is generated by the subgroups X′ and A′; note that then we have the conjugation action θ′ : A′ → Aut(X′) given by θ′(a′)(x′) = a′x′a′^{−1} for all a′ ∈ A′ and x′ ∈ X′. Let us now call this the internal semidirect product, and let us generalize it to the concept of the external semidirect product X^{∗} ⋊ A^{∗} of any group X^{∗} and any group A^{∗} relative to a given action of A^{∗} on X^{∗}, i.e., a homomorphism θ^{∗} : A^{∗} → Aut(X^{∗}). As a set, X^{∗} ⋊ A^{∗} is simply the set of all pairs (b^{∗}, a^{∗}) with b^{∗} ∈ X^{∗} and a^{∗} ∈ A^{∗}. We make this into a group by defining the product of any two pairs by the equation (b^{∗}, a^{∗})(b^{∗∗}, a^{∗∗}) = (b^{∗}θ^{∗}(a^{∗})(b^{∗∗}), a^{∗}a^{∗∗}) for all b^{∗}, b^{∗∗} in X^{∗} and a^{∗}, a^{∗∗} in A^{∗}. Let \overline{X} be the set of all pairs (b^{∗}, 1) with b^{∗} ∈ X^{∗}, and let \overline{A} be the set of all pairs (1, a^{∗}) with a^{∗} ∈ A^{∗}. Then X^{∗} ⋊ A^{∗} is easily seen to be the internal semidirect product of \overline{X} and \overline{A}. Also the action θ^{∗} becomes the conjugation action, i.e., for all b^{∗} ∈ X^{∗} and a^{∗} ∈ A^{∗} we have (1, a^{∗})(b^{∗}, 1)(1, a^{∗})^{−1} = (θ^{∗}(a^{∗})(b^{∗}), 1). For any (b^{∗}, a^{∗}) ∈ X^{∗} ⋊ A^{∗} let Φ_{a^{∗},b^{∗}}^{∗} ∈ Sym(X) be defined by putting Φ_{a^{∗},b^{∗}}^{∗}(x^{∗}) = b^{∗}θ^{∗}(a^{∗})(x^{∗}). Then (b^{∗}, a^{∗}) ↦ Φ_{a^{∗},b^{∗}}^{∗} gives a homomorphism Φ^{∗} : X^{∗} ⋊ A^{∗} → Sym(X^{∗}) and for its image we have Φ^{∗}(X^{∗} ⋊ A^{∗}) = H(X, A) with X = X^{∗} and A = θ^{∗}(A^{∗}), and for any (b^{∗}, a^{∗}) ∈ X^{∗} ⋊ A^{∗} upon letting b = b^{∗} and a = θ^{∗}(a^{∗}) we have Φ_{a^{∗},b^{∗}}^{∗} = Φ_{a,b} with Φ_{a,b} as in the Section on Analytic Geometry. Note that if A^{∗} acts faithfully on X^{∗}, i.e., if for the kernel of θ^{∗} we have ker θ^{∗} = 1, then also ker Φ^{∗} = 1, and hence in this case X^{∗} ⋊ A^{∗} is isomorphic to the internal semidirect product of H(X^{∗}, 1) and θ^{∗}(A^{∗}). On the other hand if the action of A^{∗} on X^{∗} is trivial, i.e., if for the image of θ^{∗} we have im θ^{∗} = 1, then X^{∗} ⋊A^{∗} is simply the (external) direct product of the groups X^{∗} and A^{∗}.

Remark 8.6 Let V be a finite dimensional vector space over a field k. Recall that the general linear group GL(V) is the group of all nonsingular linear transformations of V, and we defined the affine general linear group AGL(V) as an internal semidirect product by putting AGL(V) = H(V, GL(V)). Alternatively, since GL(V) < Aut(V), we may think of AGL(V) as the external direct product V ⋊ GL(V). At any rate, as said above, the two are isomorphic. Let the dimension of V be m, let us fix a basis of V, and let as usual GL(m, k) denote the group of all m × m nonsingular matrices with entries in k which we regard as acting on the column vectors by multiplying them from the left by members of GL(m, k). Then to every a^{∗} in GL(V) there corresponds a unique a in GL(m, k) and this gives an isomorphism between GL(V) and GL(m, k); note that, under this isomorphism, if a column vector x corresponds to x^{∗} ∈ V then the column vector ax corresponds to a^{∗}(x^{∗}) ∈ V. Extending this further, to every (b^{∗}, a^{∗}) in V ⋊ GL(V) there corresponds the m×(m+1) matrix (a, b) where b is the column vector which corresponds to b^{∗}, and then ax + b is the column vector which corresponds to Φ_{a^{∗},b^{∗}}^{∗}(x^{∗}) with Φ_{a^{∗},b^{∗}}^{∗} as in (8.5). This gives a group structure on the set AGL(m, k) of all m × (m + 1) matrices (a, b) over k with det(a) ≠ 0, which makes it isomorphic to V ⋊ GL(V). Note that, under this group structure, for any (a, b) and (â, b̂) in AGL(m, k) we have (a, b)(â, b̂) = (aâ, ab̂ + b), and AGL(m, k) acts on column vectors by the rule (a, b)x = ax + b. Finally it remains to note that in case k is the Galois Field GF(q) with q elements, where q is power of a prime p, it is customary to write GL(m, q) and AGL(m, q) for GL(m, k) and AGL(m, k) respectively.

9 Hilbert Revisited

Amongst the 23 problems which Hilbert formulated at the turn of the last century [27], the 13th problem asks if every function of d variables is composed of functions of d − 1 variables, with the expectation that this is not so for any d ≥ 2. In his last mathematical paper [28], published in 1927, Hilbert discussed the 13th problem in great detail by generalizing Galois’ idea of solvability thus.

Consider a polynomial

F (Y) = Y^{n} + a_{1}Y^{n−1} + ⋯ + a_{n}

of degree n with coefficients a_{1}, …, a_{n} in a field K, and assume that it has distinct roots α_{1}, …, α_{n} in an overfield of K. To this Galois associated a permutation group Gal(F, K) < S_{n} thus. Let L = K(α_{1}, …, α_{n}) be the splitting field of F over K, let Gal(L, K) be the group of all K-automorphisms of L, i.e., those automorphisms of L which keep K elementwise fixed. Then every member of Gal(L, K) permutes the roots α_{1}, …, α_{n} and this makes Gal(L, K) isomorphic to a subgroup of S_{n} which we call Gal(F, K). Galois showed that the equation F = 0 is solvable by radicals iff Gal(F, K) is solvable as a group. Here the solvability of a finite group G, as introduced by Galois, means there is a finite sequence 1 = G_{0} ◁ G_{1} ◁ ⋯ ◁ G_{e} = G such that the quotient group G_{i}/G_{i−1} is cyclic for 1 ≤ i ≤ e. Assuming K to contain all the relevant roots of 1, the solvability of the equation means that the field L can be obtained from K by successively adjoining roots of equations of the form Y^{ν} −u = 0, with the understanding that in case of positive characteristic p we must also throw in equations of the form Y^{p} − Y − u = 0. Both these are special cases of a category of equations F = 0 for which the coefficients a_{1}, …, a_{n} are functions of a single parameter u over a ground field k. This is Hilbert’s generalization of Galois’ idea of solvability. In [28], Hilbert showed that although a general quintic equation, i.e., an equation F = 0 with n = 5, is unsolvable in Galois’ sense, by Tschirnhausen transformations, i.e, generalized completing-the-square type transformations, its roots can be found by successively adjoining roots of one-parameter equations. Then he proceeded to do the same thing for sextics, i.e., equations F = 0 with n = 6, and showed that by Tschirnhausen transformations their roots can be found by successively adjoining roots of two-parameter equations, and conjectured that it is not possible to go down to one-parameter equations. This may be called his Sextic Conjecture.

In my 1997 paper [6] I showed that an affirmative answer to the Sextic Conjecture would provide an example of a function of two variables which is not composed of functions of one variable, thereby settling the d = 2 case of the 13th problem. In the same paper I also showed that to settle the Sextic Conjecture affirmatively it suffices to find a sextic F whose coefficients a_{1}, …, a_{6} belong to the two variable meromorphic series field K = k((X, Z)) = the quotient field of the (formal) power series ring k[[X, Z]] over an algebraically closed ground field k, and whose Galois group Gal(F, K) is unsolvable.

In my 1955 Ph.D. Thesis [1], I wrote down the n = 6 (i.e., m = 2 and q = 5) case of the surface covering

F (Y) = Y^{n} + XY + Z with n = 1 + q + q^{2} + ⋯ + q^{m−1}

where m > 1 is an integer and q > 1 is a power of the positive characteristic p of k, and I considered the local Galois group G = Gal(F, K). Note that F − Y F_{Y} = Z and hence the branch locus of the projection of the surface F = 0 onto the (X, Z)- plane is the line Z = 0 which of course has a simple point at the origin. The reason for constructing such a surface covering was the observation that a basic fact, which underlies Jung’s 1908 desingularization proof, says that in case of zero characteristic the local Galois group G above a simple point of the branch locus is cyclic. It was suggested by my Guru Zariski to see if a similar thing was true in nonzero characteristic, and if so then whether it could be used for resolving surface singularities in nonzero characteristic. In [1], by direct calculation I showed that for m = 2 and q = 5 the Galois group G is unsolvable. Then, in my 1994–97 papers [3] to [5], I showed that for general values of m and q we have G = PGL(m, q), where the projective general group PGL(m, q) is the quotient of the general linear group GL(m, q) by the group of all scalar matrices. A major step in the proof of this is that G is two-transitive because not only is F irreducible, but it remains so after we “throw away” a root α of it, i.e., its “twisted derivative” F′(Y) = \frac{F (Y)−F (α)}{Y −α} is also irreducible; note that the irreducibility of F is over K, but the irreducibility of F′ is over the root field K(α); see my 1992 paper [2] where the throwing away root method was introduced. This two-transitivity together with Theorem I from Cameron-Kantor’s 1979 paper [16] shows that G = PGL(m, q).

But for Hilbert’s Sextic Conjecture we only need to know that G is unsolvable. This indeed follows from Burnside’s Theorem, because the degree n of F is not a prime power, and hence the unique minimal normal subgroup of G cannot be elementary abelian, and therefore it must be nonabelian simple, and so G must be unsolvable.

It may be observed that the d > 2 case of Hilbert’s Thirteenth Problem remains quite open. For some thoughts on it, in addition to my 1997 paper [6], see the 1976 Arnold-Shimura paper [10].

Note 9.1 Cayley’s Theorem brings to the fore the amazing fact that S_{n} contains all groups of order at most n, indeed each of them having at least two copies inside S_{n}. This is parallel to the equally amazing fact, encountered in linear algebra, that the n × n matrix ring M_{n} over a field K contains inside it several copies of all separable field extensions K′ of K of degree at most n. Indeed, assuming the degree of L over K to be n, and letting F (Y) = Y^{n} + a_{1}Y^{n−1} + ⋯ + a_{n} to be the minimal monic polynomial of a primitive element of L over K, let C be the companion matrix of F, i.e., C = (C_{ij}) ∈ M_{n} is given by

C_{ij} = \begin{cases}−a_{i−1} & if 1 ≤ i ≤ n and j = n \\ 1 & if i = j + 1 and 1 ≤ j ≤ n − 1 \\ 0 & otherwise. \end{cases}

Then the subring K[C] of M_{n} is isomorphic to L. Thus S_{n} and M_{n} are very rich objects worthy of intense study. Moreover S_{n} is a subgroup of GL_{n} = GL(n, K) which itself is the group of units of the ring M_{n}. Thus the matrix ring really wins out! And since it subsumes the solutions of all equations over k of degree at most n, it too deserves to occupy a central position in Theory of Equations, which is the old name for Algebra.

Note 9.2 As hinted above, the irreducibility of the polynomial F = Y^{n} + a_{1}Y^{n−1} + ⋯ + a_{n} over K is equivalent to the transitivity of Gal(F, K), and this together with the irreducibility of its twisted derivative F′ over its root field K(α) is equivalent to the two-transitivity of Gal(F, K). It can also be seen that Gal(F, K) is primitive iff F is irreducible over K and there is no field between K and K(α). These are illustrations of Galois’ wonderful discovery that Group Theory and Theory of Equations are completely parallel disciplines, or two faces of the same fundamental reality.

10 Acknowledgements

It is a great pleasure to thank Peter Cameron, Chris Christensen, Jan Saxl, and John Thompson for stimulating discussions concerning the material of this paper.

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