PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 126, Number 5, May 1998, Pages 1337–1343 S 0002-9939(98)04491-8

THE PRIMITIVE p-FROBENIUS GROUPS

P. FLEISCHMANN, W. LEMPKEN, AND PHAM HUU TIEP

(Communicated by Ronald M. Solomon)

Abstract. Let p be a fixed prime. A finite primitive permutation group G with every two-point stabilizer G_{α,β} being a p-group is called a primitive p-Frobenius group. Using our earlier results on p-intersection subgroups, we give a complete classification of the primitive p-Frobenius groups.

1. Introduction

Let Σ_{Ω} denote the symmetric group on a finite set Ω and G ≤ Σ_{Ω} a transitive permutation group. Suppose that every two-point stabilizer G_{α,β} is trivial; then a classical result of Frobenius states that G is a semidirect product K : G_{α}, where the normal subgroup K consists precisely of the identity together with all elements g ∈ G that do not fix any point of Ω. Moreover the point stabilizer G_{α} acts on K (via conjugation) in such a way that no nontrivial element g ∈ G_{α} has a nontrivial fixed point in K. A permutation group with these properties is called a Frobenius group with Frobenius kernel K.

As a consequence of a celebrated theorem of J. G. Thompson, the Frobenius kernel K is nilpotent and G_{α} is a semiregular group, i.e. G_{α} has a faithful action on a suitable vector space V such that C_{V} (g) = 0 for every g ∈ G_{α}\{id} (notice that V can be chosen as a characteristically simple subgroup of a Sylow group of K). If moreover G is primitive, then K ≅ ℤ_{p}^{ℓ} =: V and G_{α} acts semiregularly on V.

In the course of his investigation of finite near fields, Zassenhaus [9] obtained a complete classification of finite semiregular groups. Putting all these results together, one has a complete picture of primitive Frobenius groups.

In this paper we finish our investigation of the following p-local variant of the situation above:

Definition 1.1. Let p be a fixed prime. We define F(p) to be the set of all (faithful) finite primitive permutation groups with every 2-point stabilizer G_{α,β} being a pgroup. An element of F(p) will be called a primitive p-Frobenius group.

Furthermore, F^{a}(p) denotes the subset of F(p) consisting of those groups with abelian socle, and we set F^{na}(p) := F(p)\F^{a}(p).

In [3], we defined a proper subgroup X < G to be a p-intersection subgroup of G if and only if X is not a p-group but for any g ∈ G\X the intersection X ∩ X^{g} is a p-group. The set of p-intersection subgroups of G will be denoted by I_{p}(G). The

\begin{matrix}c \\ ◯ \end{matrix}1998 American Mathematical Society

Received by the editors June 19, 1996 and, in revised form, November 5, 1996.

1991 Mathematics Subject Classification. Primary 20B15.

main result of [3] is a classification of the elements in I_{p}(G) for all almost simple groups. Now we will apply this result to obtain a complete classification of the primitive p-Frobenius groups.

Remarks. (i) It is clear that F(p) includes the class F^{∞} of all finite primitive Frobenius groups, which is the intersection ⋂_{p prime} F(p).

(ii) Every finite group G with O_{p}(G) = 1 admits a faithful transitive permutation representation with every G_{α,β} being a p-group: take the action of G on G/P, P ∈ Syl_{p}(G). Thus it is the primitivity condition that makes the class F(p) interesting.

(iii) It is immediate from the definition, that G ∈ F(p) implies that either G_{α} is a p-group or G_{α} ∈ I_{p}(G).

Our notation will be as follows: Ω always denotes a finite set and G a finite group with π(G) the set of prime divisors of |G|. For α ∈ Ω and G ≤ Σ_{Ω} we write G_{α} for Stab_{G}(α). For any subgroup S ≤ G we set Aut_{G}(S) := N_{G}(S)/C_{G}(S). Also, X < ·Y means that X is a maximal subgroup of a group Y. The socle of G, soc(G), is the product of all minimal normal subgroups of G. A subgroup G ≤ Σ_{Ω} is called primitive (resp. regular) if it is transitive and G_{α} < ·G (resp. G_{α} = 1). The symmetric, resp. alternating, group on n symbols is denoted by Σ_{n}, resp. A_{n}.

Observe that if G ∈ F^{a}(p) then soc(G) = V = 𝔽_{ℓ}^{n} is elementary abelian (the prime ℓ may differ from p) and G is the semi-direct product of V and any point stabilizer G_{α}, a ∈ Ω. In this case, one can identify Ω with V, a ∈ Ω with the zero vector of V, then embed G_{0} in GL(V) so that the action of G_{α} = G_{0} on Ω and the linear action of G_{0} on V are compatible.

Let us recall the following definition of [2]: A pair (G, V) consisting of a finite group G and a finite-dimensional 𝔽 G-module V over some field 𝔽 is called p′- semiregular if every nontrivial p′-element of G acts without any fixed points on V \ {0}. G is called p′-semiregular if (G, V) is p′-semiregular for a suitable V.

Now the following statement is immediate:

Proposition 1.2. Suppose soc(G) = V is elementary abelian. Then G ∈ F^{a}(p) if and only if the pair (G_{0}, V) is p′-semiregular, and V is a faithful irreducible G_{0}-module. ∘

In [2] all p′-semiregular pairs (G_{0}, V) have been determined.

The present paper provides a proof of the following complete classification of the groups in F(p):

Theorem 1.3. Let G be an element of F(p) and put S := soc(G). Then precisely one of the following three cases occurs:

(i) G has a regular normal subgroup V. In this case V = S is elementary abelian, G ∈ F^{a}(p) and G = V : G_{0} with p′-semiregular pair (G_{0}, V) as described in [2] (and V is a faithful irreducible G_{0}-module).

(ii) G has no regular normal subgroup and G_{α} is nilpotent. In this case p = 2, G ∈ F^{na}(2), G_{α} ∈ Syl_{2}(G) and S = O^{2}(G) = S_{1} × ⋯ × S_{k} is the unique minimal normal subgroup of G; moreover, S_{1} ≅ ⋯ ≅ S_{k} ≅ L_{2}(q) with q = 2^{n} ± 1 > 5 a prime or q = 9. Furthermore, [G : S] and k are powers of 2.

(iii) G has no regular normal subgroup and a point stabilizer G_{α} is an element of I_{p}(G). In this case S is simple, i.e. G is almost simple. Furthermore, the tuples (S, G, p, G_{α}) are as listed in Table I below.

THE PRIMITIVE p-FROBENIUS GROUPS 1339

Table I. p-intersection maximal subgroups in almost simple groups

S  G  p  G_{α}
A_{5}  G = S  3  A_{4}
A_{5}  A_{5} ≤ G ≤ Σ_{5}  2  N_{G}(ℤ_{3})
A_{6}  \begin{matrix}A_{6} ≤ G \\ ≤ Aut(A_{6}) \end{matrix}  2  N_{G}(3^{2})
A_{6}  \begin{matrix}P GL_{2}(9), M_{10}, \\ Aut(A_{6}) \end{matrix}  2  N_{G}(ℤ_{5})
A_{n}, n prime ∉ \{7, 11, 17, 23\} G = S  p with \frac{n−1}{2} = p^{f} ℤ_{n} : ℤ_{\frac{n−1}{2}}
A_{n}, 5 ≤ n ∈ F  Σ_{n}  2  ℤ_{n} : ℤ_{n−1}
L_{2}(7) ≅ L_{3}(2)  G = S  2  Σ_{4}^{(i)}, i = 1, 2
G = P GL_{2}(7)  2  N (X), N(T_{1})
≅ D_{12}
L_{2}(7)  G = S  3  B
L_{2}(11)  G = S  2  N(T_{cox})
P GL_{2}(11)  2  N(T_{1}) ≅ D_{20}
N(T_{cox}) ≅ D_{24}
L_{2}(11)  G = S  5  B
L_{2}(2^{a}), 2^{a} ≥ 4  G = S  p = 2^{a} − 1 ∈ M B
L_{2}(q), q ∉ F ∪ {4, 7, 9, 11}  π(G/S) ⊆ {2}  2  N(T_{1})
L_{2}(q), q ∉ M ∪ {4, 5, 9, 11}  π(G/S) ⊆ {2}  2  N(T_{cox})
L_{2}(2^{ℓ})  \begin{matrix}2 < |G/S| \\ = ℓ prime \end{matrix}  2  N(T_{cox})
L_{2}(r), r = 2p^{x} + 1 a prime  G = S  p > 2  B
L_{2}(3^{m}), \{ \begin{matrix}3^{m} = 2p^{x} + 1 \\ m|(p − 1) \\ m an odd prime \end{matrix}  G = S  p > 2  B
L_{2}(r), r ∈ F  G/S ≤ ℤ_{2}  2  B
P SU_{3}(q), q ∈ {3, 5, 9}  G/S ≤ ℤ_{2}  2  B
2G_{2}(3)′ ≅ L_{2}(8)  G/S ≤ ℤ_{3}  2  B, N_{G}(B)
2B_{2}(q), q = 2^{l}, l = 2a + 1 > 1 G = S  p = 2^{l} − 1 ∈ M B
G = S 2 D_{2(q−1)},

X_{±} =ℤ_{q±2\sqrt{q}+1}: ℤ_{4}

|G/S| = l prime  2  X_{±}: ℤ_{l} with 5∥X_{±}|
L_{3}(4)  G/S ≤ F ≅ ℤ_{2}  2  N(ℤ_{3}^{2} : Q_{8})
L_{3}(4)  \begin{matrix}Δ ≤ G/S \\ ≤ ℤ_{2} × ℤ_{2} \end{matrix}  2  N(ℤ_{3}^{2} : Q_{8})
3D_{4}(q)  π(G/S) ⊆ \{2\}  2  \begin{matrix}N(T_{5}), \\ |T_{5}| = q^{4} − q^{2} + 1 \end{matrix}
L_{p}(q), P SU_{p}(q)  π(G/S) ⊆ {p}  p  N(T_{cox})
M_{11}  G = S  2  3^{2} : Q_{8} · 2
J_{1}  G = S  2  Σ_{3} × D_{10}
M_{23}  G = S  11  23 : 11
BM  G = S  23  47 : 23
M (??)  G = S  29  59 : 29

Conversely, each of the groups mentioned in (i), (ii) and (iii) is a member of F(p), where in case (ii) we have to assume in addition that Aut_{G}(S_{i}) ≅ P GL_{2}(7) if q = 7 and Aut_{G}(S_{i}) ≅ P GL_{2}(9), M_{10} or Aut(A_{6}) if q = 9.

In Table I B denotes a Borel subgroup; T_{1}, T_{cox} denote split- and Coxeter tori, respectively, L_{2}(q) := P SL_{2}(q), N(H) := N_{G}(H); if G > S, N (X) := {N_{G}(X) | X ∈ I_{p}(S), maximal in S}. Here ?? means either 59 : 29 < L_{2}(59) < · M or 59 : 29 < · M. The existence of L_{2}(59) in M is not settled yet. F, resp. M, denotes the set of Fermat, resp. Mersenne, primes.

Due to the isomorphisms A_{5} ≅ L_{2}(4) ≅ L_{2}(5) and A_{6} ≅ L_{2}(9) these groups and their automorphic decorations are listed only in the alternating groups’ section. For L_{3}(4), F (resp. Δ ≅ ℤ_{2}) is generated by field (resp. graph) automorphisms.

Note that for any X ∈ I_{p}(G) there is a subgroup H ≤ G such that X ∈ I_{p}(H) and X is maximal in H. Hence the results above give also a complete classification of p-intersection subgroups occuring in arbitrary finite groups. In particular we obtain the following result which can be viewed as the p-local version of Zassenhaus’ classification of Frobenius complements:

Theorem 1.4. Let G be a finite group, p a prime and X < G a p-intersection subgroup. Then either X is p′-semiregular or X is a solvable group (occurring as G_{α} in Table I up to a suitable normal p-subgroup of X).

2. Prerequisites

The reader is referred to [2] for a complete list of p′-semiregular groups; here we restrict ourselves to listing the perfect ones. R is the set of all primes r such that r = 2^{a} · 3^{b} + 1 for a ≥ 2, b ≥ 0, and (r + 1)/2 is a prime.

R. Guralnick and R. Wiegand [5] also classified p′-semiregular groups G such that the underlying field of the corresponding G-modules has characteristic p, and pointed out a very interesting connection of these groups with multiplicative structures of Galois field extensions. The authors’ proof in [2] is independent of that in [5] and in addition also describes the relevant G-modules.

Theorem 2.1. Let G be a perfect finite group and (G, V) a p′-semiregular pair for a faithful irreducible 𝔽 G-module V. Then one of the following holds:

(i) G ≅ SL_{2}(p^{a}) for some a ≥ 1 with p^{a} > 3.

(ii) G ≅ ^{2}B_{2}(2^{2a+1}) for some a ≥ 1 with p = 2.

(iii) G ≅ ^{2}B_{2}(2^{2a+1}) × SL_{2}(2^{2b+1}) with a, b ≥ 1, gcd(2a + 1, 2b + 1) = 1 and p = 2.

(iv) G ≅ SL_{2}(r) with r ∈ R ∪ {7, 17} and p = 3.

(v) G ≅ SL_{2}(5) and p ≥ 7.

(vi) G = ES, where E = O_{2}(G) ≅ 2_{−}^{1+4}, S ≅ SL_{2}(5), E ∩ S = Z(G) ≅ ℤ_{2} and p = 2.

Conversely, if (G, p) satisfies any of the conditions (i) – (vi), then there exists a faithful absolutely irreducible G-module V such that (G, V) is p′-semiregular.

Proof. Let (G, V) be p′-semiregular for a faithful irreducible 𝔽 G-module V and char 𝔽 = ℓ. First suppose that ℓ divides |G|. Then it is clear that ℓ = p and the irreducibility of V forces O_{p}(G) = 1. By Theorem 4.1 of [2] (cf. also [5]), (G, p) is as listed in (i) – (iv). Next suppose that ℓ does not divide |G|. Then Theorem 5.6 of [2] and the irreducibility of V force (G, p) to satisfy one of the conditions (iv) – (vi) or (i) with p^{a} = 4, 5, 9. The existence of p′-semiregular pairs for the groups G listed has also been established in [2]. ∘

Example 2.2. (i) (Zassenhaus) ℓ^{2} : SL_{2}(5) ∈ F^{∞} for all primes ℓ ≡ ±1(mod10) and ℓ^{4} : SL_{2}(5) ∈ F^{∞} for all primes ℓ ≡ ±3(mod10).

(ii) p^{2a} : SL_{2}(p^{a}) ∈ F(p) for any prime p.

(iii) 3^{6} : SL_{2}(13) ∈ F(3) (Hering’s group).

(iv) 7^{8} : (2_{−}^{1+4}\A_{5}) ∈ F(2) and 7^{4} : SL_{2}(9), 5^{12} : SL_{2}(13) ∈ F(3).

The following result will be used in the next section.

Theorem 2.3. Let G be a finite group with a nilpotent maximal subgroup H. (i) (Thompson; see [4], Thm. 10.3.2) If H has odd order then G is solvable.

(ii) (Baumann; see [1]) If G is non-solvable then O^{2}(G/F (G)) is a direct product of simple groups isomorphic to L_{2}(q) with primes q of the form 2^{n} ± 1 or q = 9.

We will also need the following result, which is an easy consequence of the classification of finite simple groups:

Lemma 2.4. Let E be a nonabelian finite simple group and let α ∈ Aut(E) be an element whose order is coprime to |E|. Then C_{E}(α) is not nilpotent.

Proof. See [3]. ∘

3. Reduction to the simple socle case

Proposition 3.1. Let G ∈ F^{na}(p). Then G does not contain any regular normal subgroup. In particular, S := soc(G) is the unique minimal normal subgroup of G, C_{G}(S) = 1, and G/S ≅ G_{α}/S_{α}.

Moreover, for any point stabilizer G_{α} one of the following is true:

(i) p = 2 and G_{α} ∈ Syl_{2}(G);

(ii) G_{α} ∈ I_{p}(G).

Proof. 1) Suppose first that G_{α} is a p-group. As G_{α} is maximal and nilpotent, 2.3 implies p = 2 and so G_{α} ∈ Syl_{2}(G).

2) Suppose next that 1 ≠ R◁G with R∩G_{α} = 1. We can assume R ≤ soc(G) and R is not solvable (since G ∈ F^{na}(p)). Suppose in addition that G_{α} is not a p-group. Then we can find x ∈ G_{α} of prime order q ≠ p. Now for any 1 ≠ y ∈ R, G_{α} ∩ G_{y(α)} is a p-group. In particular, x^{y} ∉ G_{α}, so y^{x} ≠ y and x acts fixed-point-freely on R\{1}. By a well-known theorem of Thompson (cf. [4], Thm. 10.2.1) R must be nilpotent, a contradiction. Thus G_{α} must be a p-group. By 1), G_{α} ∈ Syl_{2}(G), and |R| = [G : G_{α}] is odd. So R (and G = R · G_{α}) is solvable, again a contradiction.

3) The claims concerning S now follow immediately. Furthermore, if G_{α} is a p-group, then (i) is fulfilled due to 1); otherwise one arrives at (ii). ∘

Corollary 3.2. Let G ∈ F(p). Then G ∈ F^{a}(p) if and only if G contains a regular normal subgroup. ∘

Proposition 3.3. Suppose that G ∈ F^{na}(p) and soc(G) is not simple. Then conclusion (ii) of Theorem 1.3 holds.

Proof. A basic tool for studying finite permutation groups is the reduction theorem first stated by O’Nan and Scott (see [8]). Here we are using an expanded version of this theorem given in [7]. Because of 3.2, the primitive permutation group G under question has no regular normal subgroups (and soc(G) is not simple). In this case, the O’Nan-Scott theorem says that G is either a simple diagonal action or a product action group; cf. [7]. We shall use the notation given there. In particular, B = soc(G) = S_{1} × … × S_{k} with S_{1} ≅ … ≅ S_{k} ≅ T for a non-abelian finite simple group T.

1) Suppose G is a simple diagonal action group, that is, case III(a) of [7] occurs. Then for some α ∈ Ω one has B_{α} = {(a, a, …, a) | a ∈ T }. In particular, taking g := (a, a, …, a), h := (a, 1, …, 1) for a non-identity p′-element a ∈ T, one sees that g = g^{h} ∈ B_{α} ∩ B_{h(α)}. Meanwhile h ∉ B_{α}, contrary to the assumption that G ∈ F(p).

We have shown that G is a product action group, i.e., case III(b) of [7] occurs. In this case, Ω can be identified with Γ^{ℓ} for some finite set Γ and some ℓ dividing k. Furthermore, if H denotes Aut_{G}(S_{1} × S_{2} × … × S_{k/ℓ}) (after suitably reindexing the S_{i}’s if necessary), then H acts primitively on Γ, with socle K isomorphic to T^{k/ℓ}. Moreover, H is of type II or III(a) (in the notation of [7]). Finally, G can be embedded in W = H ≀ Σ_{ℓ}, and the action of G on Ω is induced by the natural product action of W on Ω (cf. [7]).

2) Next we consider the situation when H is of type III(a). Then ℓ < k. Arguing as in 1), one sees that K_{γ} ∩ K_{γ′} is not a p-group for some γ, γ′ ∈ Γ, γ ≠ γ′. For the distinct points α := (γ, …, γ), α′ := (γ′, …, γ′) in Ω, one has B_{α} = (K_{γ})^{ℓ}, B_{α′} = (K_{γ′})^{ℓ}. In particular, B_{α} ∩ B_{α′} is not a p-group, a contradiction.

Thus H must be of type II, i.e., k = ℓ.

3) At this point we show that H_{γ} is a p-group.

First observe that B_{α} is a p-group, with α = (γ, …, γ). Indeed, suppose s = (s_{1}, ..., s_{k}) ∈ B_{α} is not a p-element. Without loss we may suppose s_{1} ∈ T is not a p-element. Then s = s^{t} ∈ B_{α} ∩ B_{t(α)} with t := (s_{1}, 1, …, 1). This implies that t ∈ B_{α}. Now taking u := (1, g_{2}, …, g_{k}), one has t = t^{u} ∈ B_{α} ∩ B_{u(α)} for any g_{i} ∈ T. We conclude that all (1, g_{2}, …, g_{k}) are contained in B_{α}, contradicting the equality B_{α} = (K_{γ})^{k}.

Suppose there is an element x ∈ H_{γ} whose order is a prime r different from p. Every element in W can be canonically written in the form (g_{1}, …, g_{k})π for g_{i} ∈ Σ_{Γ} and π ∈ Σ_{k}. Then G_{α} contains an element g = (x, g_{2}, …, g_{k})π with π(1) = 1. For any y ∈ C_{K}(x) and h := (y, 1, …, 1), we have g = g^{h} ∈ G_{α} ∩ G_{h(α)}, yielding h ∈ B_{α}. Due to the above observation C_{K}(x) is a p-group. Choose Q ∈ Syl_{r}(H) with x ∈ Q. Then Q∩K ◁Q and Z(Q)∩(Q∩K) ≤ C_{K}(x). So 1 = Z(Q)∩(Q∩K), which implies that 1 = Q ∩ K. But Q ∩ K ∈ Syl_{r}(K), hence (r, |K|) = 1. Now 2.4 applied to K and x provides a contradiction.

4) We have proved that H_{γ} is a p-group. If p is odd, then the maximality of H_{γ} in H together with 2.3 implies that H is solvable, a contradiction. So we conclude that p = 2. We claim that G_{α} is a 2-group. For, suppose that g = (g_{1}, …, g_{k})π ∈ G_{α} has order r, an odd prime, and α = (γ, …, γ). Observe that G_{α} ≤ W_{α} = H_{γ} ≀Σ_{k}. Since g_{i} ∈ H_{γ} has order a power of 2, we conclude that π has order r. In particular, we may suppose that π permutes the groups S_{1}, …, S_{r} cyclically. Choose c ∈ K\K_{γ}. Then for y := (c, 1, …, 1) ∈ B we have [y^{g^{i}}, y^{g^{j}}] = 1 for all i, j = 1, 2, …, r. Hence ỹ:= yy^{g}y^{g^{2}}...y^{g^{r−1}} ∈ C_{B}(g). From this it follows that g = gỹ ∈ G_{α} ∩ G_{ỹ(α)}, yielding ỹ ∈ B_{α}. But in this case c belongs to K_{γ}, contrary to the choice of c. Consequently, G_{α} is a 2-group, and so G_{α} ∈ Syl_{2}(G).

Applying 2.3 to the maximal subgroup G_{α} of G, we come to conclusion (ii) of Theorem 1.3. ∘

The following is one of the main results in [3]; here it classifies the elements in F^{na}(p) with simple socle:

Theorem 3.4. Let S ⊴ G ≤ Aut(S) with S a finite nonabelian simple group, and suppose that X ∈ I_{p}(G) is maximal in G. Then (S, G, p, X = G_{α}) is one of the tuples listed in Table I.

Proof. See [3]. ∘

4. Proof of Theorem 1.3

By 3.2 and the remarks in the introduction we can assume that G ∈ F^{na}(p). First we suppose that G_{α} is nilpotent. Applying 2.3, we see that F (G) = 1 (as

G ∈ F^{na}(p)) and S := soc(G) = O^{2}(G) is the direct product of k isomorphic (since S is characteristically simple) simple groups S_{1} ≅ … ≅ S_{k} ≅ L_{2}(q) with q = 2^{n} ± 1 a prime or q = 9. If k = 1, then a direct inspection of maximal subgroups of G with L_{2}(q) ≤ G ≤ Aut(L_{2}(q)) shows that p = 2 and G_{α} ∈ Syl_{2}(G), that is, conclusion (ii) of 1.3 holds. If k > 1, then, applying 3.3, one again obtains that p = 2 and G_{α} ∈ Syl_{2}(G). Moreover, due to 3.1, G/S ≅ G_{α}/S_{α} and hence [G : S] is a 2-power. But G/S acts transitively on the set {S_{1}, …, S_{k}}; therefore k is a 2-power. Thus (ii) is fulfilled.

Now suppose that (ii) does not hold. Then G_{α} ∈ I_{p}(G). In this case 3.3 shows that soc(G) is simple. Applying 3.4, we arrive at (iii). ∘

Acknowledgements

The authors are grateful to Prof. R. Solomon for helpful discussions, and to the referee for his valuable comments that greatly improved the exposition of the paper.

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(P. Fleischmann and W. Lempken) Institute for Experimental Mathematics, University of Essen, Ellernstr. 29, 45326 Essen, Germany

(Pham Huu Tiep) Department of Mathematics, Ohio State University, Columbus, Ohio 43210

  E-mail address: tiep@math.ohio-state.edu