Extending Characters from Hall Subgroups

Gunter Malle and Gabriel Navarro^{1}

Received: May 8, 2011 Revised: October 10, 2011

Communicated by Wolfgang Soergel

Abstract. Suppose that G is a finite π-separable group. A classical result asserts that all irreducible characters of a Hall π-subgroup H of G extend to G if and only if H has a normal complement in G. Now, we fix a prime p and analyze when only the p′-degree irreducible characters of H extend to G.

2010 Mathematics Subject Classification: 20C15 (primary), 20G40 (secondary) Keywords and Phrases: extensions of characters, Hall subgroups, Sylow normalizers, π-separable groups

1. Introduction

We come back to an old result of C. Sah ([18]) that asserts that in a finite π-separable group, all irreducible complex characters of a Hall π-subgroup H of G extend to G if and only if G has a normal π-complement. Now, we fix a prime p and we wish to characterize when only the p′-degree characters of H extend to G.

Theorem A. Let G be a finite π-separable group. Let H be a Hall π-subgroup of G, let K be a π-complement of G, and let p be a prime. Then every α ∈ Irr(H) of p′-degree extends to G if and only if there is P ∈ Syl_{p}(H) such that N_{G}(P) ⊆ N_{G}(K).

Of course, Theorem A is far more general than Sah’s theorem, although we pay the price of using the Classification of Finite Simple Groups. This is not that surprising, however: in the case where H is normal in G, Theorem A is equivalent to proving a well-known consequence of the (yet unproven) McKay conjecture (see [17, Thm. C]) which therefore now becomes established. As an easy consequence of our main result we obtain:

1The research of the second author is partially supported by the Spanish Ministerio de Educación y Ciencia proyecto MTM2010-15296, Programa de Movilidad, and Prometeo/Generalitat Valenciana.

Corollary B. Suppose that A acts coprimely on a finite group G, and let P ∈ Syl_{p}(G) be A-invariant. Then all p′-degree irreducible characters of G are A-invariant if and only if [N_{G}(P), A] = 1.

The paper is split into two parts. In Section 2 we prove Theorem A and Corollary B modulo a statement (Theorem 2.2) on finite quasi-simple groups which is then shown in Section 3, using the classification, properties of algebraic groups and Deligne–Lusztig theory.

2. Proof of Theorem A

In our first result we use the Gajendragadkar’s π-special characters (whose main properties can be found in [3]).

Lemma 2.1. Suppose that G is π-separable and let H be a Hall π-subgroup of G. Let L ◁ G. Suppose that α ∈ Irr(H) extends to G and is such that H ∩ L ⊆ ker α. Then there is an extension β ∈ Irr(G) such that L ⊆ ker β. In particular, if all p′-degree irreducible characters of H extend to G, then all p′-degree irreducible characters of HL/L extend to G/L.

Proof. Let α̃ ∈ Irr(HL) be the unique irreducible character of HL that extends α and has L in its kernel. Notice that, by using the definition, α̃ is π-special. By hypothesis, α extends to G. By [6, Thm. F], α has a π-special extension α̂ to G. Now by [3, Prop. (6.1)], we have that α̂_{HL} is a π-special extension of α_{H} to HL. But, again by the uniqueness part in loc.cit., we have that α̂_{HL} = α̃, and we are done. □

In order to prove Theorem A, we shall need the following non-trivial result whose proof (that uses the Classification of Finite Simple Groups) we defer until Section 3 below.

Theorem 2.2. Let p be a prime. Suppose that M ◁ G has p-power index, and has a normal Hall subgroup S such that M/S is not divisible by p. Assume that S is a non-abelian quasi-simple group of order divisible by p with C_{G}(S) = Z(S) a p′-group. Let P ∈ Syl_{p}(G). Then the following are equivalent:

(i) all p′-degree P -invariant irreducible characters of S are M-invariant, and (ii) there exists a complement K of S in M normalized by P and with

[N_{S}(P), K] = 1.

We shall frequently write condition (ii) of the above theorem in the following more convenient form.

Lemma 2.3. Let S ◁ M. Suppose that P, K are subgroups of M with S ∩K = 1. Then P normalizes K and [N_{S}(P), K] = 1 if and only if N_{SP} (P) ⊆ N_{M}(K).

Proof. Since P ⊆ N_{SP} (P), by Dedekind’s lemma we have that N_{SP} (P) = N_{S}(P)P. Thus, if P normalizes K and [N_{S}(P), K] = 1 it is clear that N_{SP} (P) ⊆ N_{M}(K). Conversely, we have that P and N_{S}(P) normalize K. Then, using that S ◁ M, we have that [N_{S}(P), K] ⊆ S ∩ K = 1. □

Also, the group theoretical conclusion in Theorem A has another convenient form.

Lemma 2.4. Suppose that G is a π-separable group with a Hall π-subgroup H and a Hall π-complement K. Let P ∈ Syl_{p}(H). If P normalizes K, then N_{G}(P) = N_{H}(P)N_{K}(P). In particular, N_{G}(P) ⊆ N_{G}(K) if and only if N_{H}(P) ⊆ N_{G}(K).

Proof. This is an application of Lemma (2.1) of [10]. □

Lemma 2.5. Suppose that P is a p-group acting on a group S that does not have a normal p-complement. Then there exists 1 ≠ χ ∈ Irr(S) which is P -invariant of p′-degree.

Proof. The semidirect product SP cannot have a normal p-complement. By Thompson’s Theorem (Corollary (12.2) of [5]), there exists a non-linear γ ∈ Irr(SP) of degree not divisible by p. Now, γ_{S} = χ is irreducible (by Corollary (11.29) of [5]) and P -invariant. It is clear that χ is not trivial since otherwise γ would be a p′-degree character of a p-group, while γ is not linear. □

The following is one direction of Theorem A.

Theorem 2.6. Let G be a finite π-separable group. Let H be a Hall π-subgroup of G, let K be a π-complement of G, and let p be a prime. Suppose that every α ∈ Irr(H) of p′-degree extends to G. Then there is P ∈ Syl_{p}(H) such that N_{G}(P) ⊆ N_{G}(K).

Proof. If p does not divide |H|, then all irreducible characters of H extend to G, and G has a normal π-complement by Sah’s Theorem (see [18, Thm. 5]). So we may assume that p ∈ π. We prove that there exists P ∈ Syl_{p}(H) such that N_{G}(P) ⊆ N_{G}(K) by induction on |G|. Since H is a Hall subgroup of G, notice that every Sylow p-subgroup of H is a Sylow p-subgroup of G. Let N be a fixed but arbitrary minimal normal subgroup of G. Step 1. We can assume that there is P ∈ Syl_{p}(H) such that N_{G}(P) ⊆ N_{G}(K)N. Also, N ⊆ H. In particular, O_{π′}(G) = 1. By Lemma 2.1, we know that all p′-degree irreducible characters of HN/N extend to G/N. Therefore, by induction and using that N_{G}(KN) = N_{G}(K)N, we conclude that there is P ∈ Syl_{p}(H) such that N_{G}(P) ⊆ N_{G}(K)N. Now, since G is π-separable, we have that either N is a π-group or a π′-group. In the second case, N ⊆ K, and N_{G}(P) ⊆ N_{G}(K). So we will assume in the following that N is a π-group. Hence, N ⊆ H. Step 2. It suffices to show that there is m ∈ N such that N_{NP} (P^{m}) normalizes K. Indeed, suppose that N_{NP} (P^{m}) normalizes K for some m ∈ N. Hence P^{m}, which is contained in NP, normalizes K. Then we have by Step 1 that N_{G}(P^{m}) ⊆ N_{G}(K)N = G_{1}. Now, P^{m} ∈ Syl_{p}(G_{1}), and we may apply Lemma (2.1) of [10] to conclude that

N_{G_{1}}(P^{m}) = N_{N}(P^{m})N_{N_{G}(K)}(P^{m}) ⊆ N_{G}(K) .

However N_{G}(P^{m}) ⊆ G_{1}, so N_{G_{1}}(P^{m}) = N_{G}(P^{m}). Now, P^{m} ⊆ P N ⊆ H is a Sylow p-subgroup of H, and we are done in this case. Step 3. If θ ∈ Irr(N) of p′-degree extends to P N, then θ is K-invariant. If η ∈ Irr(P N) is such an extension, then η^{H} has p′-degree. Hence there exists ψ ∈ Irr(H) over η of p′-degree. By hypothesis, we have that ψ extends to some χ ∈ Irr(G). Let T be the stabilizer of θ in G, so that P N ⊆ T. If ν ∈ Irr(T |θ) is the Clifford correspondent of χ over θ, then ν^{G} = χ has π-degree. It follows that T contains some π-complement of G. Thus K^{g} is contained in T for some g ∈ G. Now, we know that P normalizes KN by Step 1, and hence |G : N_{G}(K)N| is not divisible by p. Thus |G : N_{G}(K^{g})N| is not divisible by p. By Corollary (1.2) of [19] applied in the group G/N with respect to the subgroup T/N, we have that |T : N_{T} (K^{g})N| divides |G : N_{G}(K^{g})N|, and therefore |T : N_{T} (K^{g})N| is not divisible by p either. It follows that there is some R ∈ Syl_{p}(T) normalizing K^{g}N. Hence R ∈ Syl_{p}(G). Now, R and P^{g} are Sylow p-subgroups of N_{G}(K^{g})N. Thus P^{gmn_{0}} = R for some m ∈ N_{G}(K^{g}) and n_{0} ∈ N. Also, R = P^{v} for some v ∈ T because R and P are Sylow psubgroups of T. Now, write m = x^{g} for some x ∈ N_{G}(K), so that gm = xg. We have that P^{xgn_{0}} = R = P^{v} and thus xgn_{0}v^{−1} ∈ N_{G}(P) ⊆ N_{G}(K)N. Hence, gn_{0}v^{−1} ∈ N_{G}(K)N, and gn_{0}v^{−1} = wn for some w ∈ N_{G}(K) and n ∈ N. Finally, since K^{g} ⊆ T, we have that K^{gn_{0}v^{−1}} ⊆ T (because n_{0}, v ∈ T). Thus K^{wn} ⊆ T and K = K^{w} ⊆ T^{n^{−1}} = T, as claimed. Step 4. We can assume that NKP = G. Thus H = NP and M = NK ◁ G. By Step 1, we have that P ⊆ N_{G}(KN) = N_{G}(K)N, so G_{0} = NKP is a subgroup of G and NK ◁ G_{0}. Write H_{0} = NP and G_{0} = NKP, and notice that H_{0} is a Hall π-subgroup of G_{0}. Also, K is a π-complement of G_{0}. Let η ∈ Irr(H_{0}) with p′-degree. Since |H_{0} : N| is a power of p, then, by Corollary (11.29) of [5], we have that θ = η_{N} ∈ Irr(N) has p′-degree. Now θ has an extension to H_{0}. By Step 3, we conclude that θ is K-invariant. Now, (|KN : N|, |N|) = 1 and therefore θ has a canonical extension θ̂ to KN by Corollary (8.16) of [5], which is by uniqueness, therefore P -invariant. Hence, by Corollary (4.2) of [6], it follows that restriction defines a bijection

Irr(G_{0}|θ̂) → Irr(H_{0}|θ) .

We conclude that η extends to G_{0}. Now suppose that G_{0} < G. Then, by induction, we conclude that there is P_{0} ∈ Syl_{p}(H_{0}) such that N_{G_{0}}(P_{0}) ⊆ N_{G_{0}}(K). Since P ∈ Syl_{p}(H_{0}), we have that P_{0} = P^{n} for some n ∈ N. Then N_{NP} (P^{n}) ⊆ N_{G_{0}}(P_{0}) ⊆ N_{G}(K), and we apply Step 2 in this case. Hence, we are reduced to the case where G_{0} = G, H = NP and M = NK ◁ G. Step 5. We can assume that N is a direct product of non-abelian simple groups of order divisible by p which are transitively permuted by G. In particular, O_{p′}(G) = O_{p}(G) = 1. Also, we can assume that every p′-degree P -invariant irreducible character of N is M-invariant. Suppose first that N is a p′-group. Hence, P acts coprimely on NK, and because NK is π-separable, it follows (using, for instance, that the number of Hall ρ-subgroups of KN is not divisible by p, where ρ is the set of prime divisors of |K|) that P normalizes some K^{n} for some n ∈ N. Now every P -invariant character of N extends to P N (by Corollary (8.16) of [5]), and by Step 3 is K-invariant. Therefore, every P -invariant character of N is NKinvariant, and therefore K^{n}-invariant. Thus every irreducible P -invariant character of N is P K^{n}-invariant and by Lemma (2.2) of [16], we conclude that C_{N}(P K^{n}) = C_{N}(P). Hence, N_{N}(P) = C_{N}(P) ⊆ C_{N}(K^{n}) ⊆ N_{G}(K^{n}). We had that P^{n^{−1}} normalizes K and now we have that N_{N}(P^{n^{−1}}) ⊆ N_{G}(K). Thus N_{NP} (P^{n^{−1}}) = N_{N}(P^{n^{−1}})P^{n^{−1}} ⊆ N_{G}(K), and this case is complete by Step 2. Suppose now that N is a p-group. Hence N ⊆ P and H = P. In this case, the hypotheses tell us that every linear character of P extends to G. By Tate’s Theorem (use, for instance, Theorem (6.31) of [5]), we conclude that G has a normal p-complement. Hence K ◁ G, and in this case the theorem is proved. So we may assume that N is a direct product of non-abelian simple groups of order divisible by p, which are transitively permuted by G. In particular O^{p}(N) = N, and it follows that every P -invariant p′-degree irreducible character of N extends to P N by Corollary (8.16) of [5]. Therefore by Step 3, every P -invariant p′-degree character of N is K-invariant, and hence M-invariant. Step 6. We can assume that N is a minimal normal subgroup of NP. Hence N = S^{g_{1}} × ⋯ × S^{g_{t}}, where {S^{g_{1}}, …, S^{g_{t}}} is the P -orbit of S, a non-abelian simple group of order divisible by p, g_{i} ∈ P, and g_{1} = 1. Also, we can assume that t > 1. We can write N = U ×V, where U > 1 and V ≥ 1 are P -invariant, and U is the direct product of the P -orbit of a simple group S. That is, U = S^{g_{1}} ×⋯×S^{g_{t}}, where {S^{g_{1}}, …, S^{g_{t}}} is the P -orbit of S, g_{i} ∈ P, and g_{1} = 1. By Lemma 2.5, let 1 ≠ η ∈ Irr(S) be N_{P} (S)-invariant of p′-degree. Then it is straightforward to show that ν = η^{g_{1}} × ⋯ × η^{g_{t}} is P -invariant. Now, let τ = ν × 1_{V} ∈ Irr(N), which is P -invariant of p′-degree. Then τ is K-invariant by Step 5. Hence ker τ < N is K-invariant, and therefore G-invariant. Since V ⊆ ker τ, we conclude that V = 1, because N is a minimal normal subgroup of G. Suppose now that N is simple. Since Z(N) = 1, we have that C_{M}(N) is a π′-group. Since by Step 1, we know that O_{π′}(G) = 1, then we have that C_{M}(N) = 1. Since G/M is a p-group, then we conclude that C_{G}(N) is a pgroup. But we know that O_{p}(G) = 1, and thus we conclude that C_{G}(N) = 1. Then, and using Step 5, we are in the hypothesis of Theorem 2.2. We conclude by this theorem (and Lemma 2.3) that there is a complement K_{1} of N in M such that N_{NP} (P) ⊆ N_{G}(K_{1}). Now, K_{1} = K^{n} for some n ∈ N, and we have that N_{NP} (P^{n^{−1}}) ⊆ N_{G}(K). Then we are done by Step 2. Step 7. Conclusion.

As before, write N = S_{1} × ⋯ × S_{t}, where S_{i} = S^{g_{i}}, g_{i} ∈ P and S = S_{1}. Also recall that t > 1 and therefore N_{G}(S) < G. By using Steps 5 and 6, we have that G = P N_{G}(S). Thus |G : N_{G}(S)| is a power of p. Now, since G/N has a normal p-complement M/N, and N ⊆ N_{G}(S), we conclude that M ⊆ N_{G}(S). In particular, M ⊆ N_{G}(S^{g_{i}}) for all i. Now, using that G/M is a p-group, let G_{2} ◁ G be containing N_{G}(S) with |G_{2}| < |G|. Notice that N_{P} (S) is a Sylow p-subgroup of N_{G}(S). Also N_{P} (S) ⊆ P_{2} = P ∩ G_{2} ∈ Syl_{p}(G_{2}). Now, let η ∈ Irr(NP_{2}) of p′-degree. We have that η_{N} ∈ Irr(N) is P_{2}-invariant of p′-degree. Write η_{N} = θ_{1} × ⋯ × θ_{t}, where θ_{i} ∈ Irr(S_{i}). Then θ_{1} is N_{P} (S)-invariant, because we can write N = S × S′, where S and S′ are N_{P} (S)-invariant. Now, let

θ = θ_{1} × (θ_{1})^{g_{2}} × ⋯ × (θ_{1})^{g_{t}} ,

which is P -invariant. By Step 5, the character θ is M-invariant. In particular θ_{1} is M-invariant. Since N_{P} (S^{g_{i}}) ⊆ P_{2} ◁ P, we can repeat the same argument with every S^{g_{i}} and every θ_{i} to conclude that η_{N} is M-invariant. By induction applied in the group G_{2} with respect to the Hall π-subgroup NP_{2} and Hall π-complement K, we conclude that there exists P_{3} ∈ Syl_{p}(P_{2}N) such that N_{G_{2}}(P_{3}) ⊆ N_{G}(K). Hence P_{3} normalizes K and is such that [N_{N}(P_{3}), K] = 1 by Lemma 2.3. Now, P_{3}∩N ∈ Syl_{p}(N), and also [P_{3}∩N, K] = 1. In particular, P_{3} ∩ N ⊆ N_{N}(K) and by the Frattini argument, we see that |G : N_{G}(K)| is not divisible by p. Now, P_{3} ⊆ P_{1} ∈ Syl_{p}(N_{G}(K)), and we have that P_{1} ∈ Syl_{p}(G). Recall that G = MP = (KN)P. Since P ∈ Syl_{p}(G), we may write (P_{1})^{kn^{−1}x} = P for some k ∈ K, x ∈ P and n ∈ N. Hence (P_{1})^{k} = P^{n}. Since P_{1} normalizes K, we have that (P_{1})^{k} normalizes K and thus P^{n} normalizes K. Also, since [N_{N}(P_{3}), K] = 1, we have that [N_{N}((P_{3})^{k}), K] = 1, where (P_{3})^{k} ⊆ P^{n}. Finally, let Q = (P_{3})^{k} ∩ N = (P_{3} ∩ N)^{k} ∈ Syl_{p}(N). Now, notice that, by elementary group theory, if R is any p-subgroup of G such that R ∩ N = Q, then N_{G}(R) ⊆ N_{G}(Q) and N_{N}(R)/Q = C_{N_{N} (Q)/Q}(R). Since (P_{3})^{k} ∩ N = Q ∈ Syl_{p}(N), and (P_{3})^{k} ⊆ (P_{1})^{k}, we also have that (P_{1})^{k} ∩ N = Q. In particular,

N_{N}((P_{1})^{k})/Q = C_{N_{N} (Q)/Q}((P_{1})^{k}) ⊆ C_{N_{N} (Q)/Q}((P_{3})^{k}) = N_{N}((P_{3})^{k})/Q

and we conclude that N_{N}((P_{1})^{k}) ⊆ N_{N}((P_{3})^{k}) ⊆ C_{G}(K). Hence, we have found (P_{1})^{k} ∈ Syl_{p}(G) such that (P_{1})^{k} normalizes K and [N_{N}((P_{1})^{k}), K] = 1. Hence

N_{N(P_{1})^{k}}((P_{1})^{k}) ⊆ N_{G}(K)

using Lemma 2.3. Since (P_{1})^{k} = P^{n} ∈ Syl_{p}(H), we use Step 2 to finish the proof of the theorem. □

In order to prove the remaining direction of Theorem A, we need one more lemma.

Lemma 2.7. Suppose that A acts as automorphisms on a finite group G, where G is a direct product of a set of subgroups X, which are permuted by A. Let B ⊆ A, and suppose that B acts transitively on X. Let S ∈ X. Then C_{G}(A) = C_{G}(B) if and only if C_{S}(N_{A}(S)) = C_{S}(N_{B}(S)).

Proof. By [9, Lemma 2.2], we have that C_{G}(A) ≅ C_{S}(N_{A}(S)) and C_{G}(B) ≅ C_{S}(N_{B}(S)). Since C_{G}(A) ⊆ C_{G}(B) and C_{S}(N_{A}(S)) ⊆ C_{S}(N_{B}(S)), the proof easily follows. □

The following completes the proof of Theorem A.

Theorem 2.8. Let G be a finite π-separable group. Let H be a Hall π-subgroup of G, let K be a π-complement of G, and let p be a prime. Suppose that there is P ∈ Syl_{p}(H) such that N_{G}(P) ⊆ N_{G}(K). Then every α ∈ Irr(H) of p′-degree extends to G.

Proof. Again, we can assume that p ∈ π. We argue by double induction, first on |G : O_{π}(G)|, and second on |G|. If U ≤ G and N ◁ G, then notice that |U : O_{π}(U)| ≤ |G : O_{π}(G)|, and that |G/N : O_{π}(G/N)| ≤ |G : O_{π}(G)|. By hypothesis, we have that K ◁ KP ≤ G. Also, if KP ≤ U < G, by induction, we have that the theorem is valid for U with respect to any Hall subgroup of U containing P. Let α ∈ Irr(H) be of p′-degree. We want to show that α extends to G. If N ◁ G, then we have that N_{G/N}(P N/N) ⊆ N_{G/N}(KN/N). Therefore, if 1 < N is a π′-group, we easily see that |G/N : O_{π}(G/N)| < |G : O_{π}(G)|, and we deduce that α̂ ∈ Irr(HN/N) (the unique extension of α to HN having N in its kernel) extends to G/N, by induction. Hence, we may assume that O_{π′}(G) = 1. Now, let N = O_{π}(G) ⊆ H. Suppose that Z ⊆ N is normal in G. Since α has p′- degree, there exists η̂ ∈ Irr(ZP) of p′-degree under α. Hence η = η̂_{Z} ∈ Irr(Z), because ZP/Z is a p-group. Now, if ZP K < G, by induction, the group ZP K with Hall subgroup ZP and complement K satisfies the hypothesis of the theorem. We conclude that η̂ extends to ZP K, and therefore that η is K-invariant. Hence P K ⊆ T = I_{G}(η), the stabilizer of η in G. We have that HT = G because K ⊆ T. Let ν ∈ Irr(T ∩H) be the Clifford correspondent of α over η. If T < G, by induction, we conclude that ν has an extension ν̂ ∈ Irr(T). Then (ν̂^{G})_{H} = ν^{H} = α (using Mackey), and we are done in this case. Hence, we conclude that whenever Z ⊆ N is normal in G then either ZKP = G or η is G-invariant. In the latter case, where η is G-invariant, we use the theory of character triple isomorphisms, as developed in [7]. Since η extends to ZP, then by using Theorem 5.2 of [7] and its proof, we can find a character triple (G^{∗}, Z^{∗}, η^{∗}) isomorphic to (G, Z, η), where Z^{∗} is a p′-group, and also a π-group. Now, (P Z)^{∗} = P^{∗} × Z^{∗} for a unique Sylow p-subgroup P^{∗} of G^{∗}. Also, H^{∗} is a Hall π-subgroup of G^{∗} and, using the Schur–Zassenhaus theorem, we have that (KZ)^{∗} = K^{∗} × Z^{∗} for a unique subgroup K^{∗} of G^{∗}, which turns out to be a Hall π-complement of G^{∗}. Also, using that N_{G}(P Z)^{∗} = N_{G^{∗}}(P^{∗} × Z^{∗}) = N_{G^{∗}}(P^{∗}) and that N_{G^{∗}}(K^{∗} × Z^{∗}) = N_{G^{∗}}(K^{∗}), we see that the hypotheses of the theorem are satisfied in G^{∗}. Furthermore, N^{∗} = O_{π}(G^{∗}), and therefore |G : O_{π}(G)| = |G^{∗} : O_{π}(G^{∗})|. Now let θ ∈ Irr(N) be an irreducible constituent of α_{N}. Suppose that θ is Ginvariant. Then, using the notation of the previous paragraph with Z = N, we have that N^{∗} = O_{π}(G^{∗}) is central in G^{∗}. By the Hall–Higman’s 1.2.3 Lemma, it follows that V = O_{π′}(G^{∗}) > 1. Then |G^{∗}/V : O_{π}(G^{∗}/V)| < |G^{∗} : N^{∗}| = |G : O_{π}(G)|, and by induction, and arguing as in the first paragraph of the proof, we conclude that α^{∗} extends to G^{∗}, and therefore that α extends to G. Hence, by the previous paragraph, we may assume that NP K = G. Thus H = NP. Suppose now that θ is K-invariant. In this case, θ has a canonical extension ρ to M = NK ◁ G, using that (|M : N|, |N|) = 1. Also ρ is P -invariant by uniqueness. Also, in this case, we know by Corollary (4.2) of [6] that restriction defines a bijection Irr(G|ρ) → Irr(NP |θ), and we conclude that α extends to G. Hence, it is enough to show that θ is K-invariant. Now, let N/Z be a chief factor of G. Since ZKP < G, we conclude by the second paragraph of the proof that α_{Z} has a G-invariant irreducible constituent. Hence, by using again character triple isomorphisms, it is no loss to assume that Z is a central p′-subgroup of G. In our present situation, and using Lemma 2.3, notice that our hypotheses now are that P normalizes K (that is, M ◁ G) and that [N_{N}(P), K] = 1. If N/Z is a p′-group, then N_{N}(P) = C_{N}(P), and thus C_{N}(KP) = C_{N}(P). In this case, and using that θ is P -invariant, θ is K-invariant by Lemma (2.2) of [16]. If N/Z is a p-group, since P normalizes K, we have that [O_{p}(G), K] = 1, and in this case we have that θ is K-invariant too (since N ⊆ O_{p}(G) × Z). Hence we may assume that N/Z is the direct product S_{1}/Z ×…×S_{t}/Z of nonabelian simple groups of order divisible by p which are transitively permuted by G. In particular O_{p}(G) = 1. Since O_{π′}(G) = 1, we easily deduce that C_{M}(N) = Z(N) = Z. Now, C_{H}(N) is a Hall π-subgroup of C_{G}(N), and C_{K}(N) is a Hall π-complement of C_{G}(N). Hence C_{G}(N) = C_{H}(N)C_{K}(N). Since C_{K}(N) = Z, we see that C_{G}(N) ⊆ H = NP. In particular, C_{G}(N)/Z is a p-group. Since O_{p}(G) = 1, we see that C_{G}(N) = Z. Now, since N/Z is a direct product of non-abelian simple groups, we have that N′Z = N. Since Z is a p′-group, then N/N′ is a p′-group, and N ∩ P = N′ ∩ P. Suppose that N is not perfect. If N′P = NP, then N = N ∩ N′P = N′(N ∩ P) ⊆ N′. Thus N′P < P. By Corollary (4.2) of [6], restriction then defines a bijection Irr(NP |λ) → Irr(N′P |λ_{Z∩N′}), and also Irr(N|λ) → Irr(N′|λ_{Z∩N′}). By the inductive hypothesis applied in N′P K, we conclude that α_{NP′} extends to N′P K. Thus α_{N′} is K-invariant. By uniqueness in the restriction map, we deduce that α_{N} = θ is also K-invariant, and we are done in this case too. Thus we may assume that N is perfect. If t = 1, that is, if N/Z is simple, then we may apply Theorem 2.2 to conclude that θ is K-invariant. So we may assume that t > 1. Hence N_{G}(S_{1}) < G. Let Q = P ∩ N ∈ Syl_{p}(N), and let Q_{i} = Q ∩ S_{i} = P ∩ S_{i} ∈ Syl_{p}(S_{i}). Since [N_{N}(P), K] = 1, we have that [Q, K] = 1. Now, let 1 ≠ x ∈ Q_{i}, and let k ∈ K. Then x = x^{k} ∈ S_{i} ∩ (S_{i})^{k}. If (S_{i})^{k} ≠ S_{i}, then (S_{i})^{k} ∩ S_{i} = Z, a p′-group, and this is not possible. We conclude that K ⊆ N_{G}(S_{i}) for all i. Since S_{i} ◁ N, then M ⊆ N_{G}(S_{i}) and we conclude that all the S_{i}’s are P -conjugate, and P N_{G}(S_{i}) = G. Now, using that Z is a p′-group, we have that Q = Q_{1} × ⋯ × Q_{t}. Write S_{i} = S^{g_{i}} for some g_{i} ∈ P, where S = S_{1}. Let P_{0} = N_{P} (S) ∈ Syl_{p}(N_{G}(S)). Since N ⊆ N_{G}(S), we have that P_{0} ∩ N ∈ Syl_{p}(N) and P_{0} ∩ S ∈ Syl_{p}(S). Necessarily, P_{0} ∩ N = Q and P_{0} ∩ S = Q_{1}. Also, since P normalizes K, then P_{0} normalizes KS. Thus KS ◁ KSP_{0} = G_{0} < G. We wish to apply the inductive hypothesis in G_{0}, where here H_{0} = SP_{0} is a Hall π-subgroup of G_{0}, and K is a π-complement of G_{0}. Notice that P_{0} ∈ Syl_{p}(SP_{0}), since P_{0} ∩ S ∈ Syl_{p}(S) and |SP_{0} : P_{0}| = |S : S ∩ P_{0}|. By Lemma 2.4, we need to check that N_{SP_{0}}(P_{0}) normalizes K. By Lemma 2.3, we need to check that P_{0} normalizes K and [N_{S}(P_{0}), K] = 1. Since P normalizes K, we only need to check that [N_{S}(P_{0}), K] = 1. By hypothesis, we know that [N_{N}(P), K] = 1. Now, P_{0} ∩ N_{S}(Q_{1}) = Q_{1} and therefore N_{G_{0}}(P_{0}) ⊆ N_{G_{0}}(Q_{1}). We easily conclude that

N_{S}(P_{0})/Q_{1} = C_{N_{S}(Q_{1})/Q_{1}}(P_{0}) .

By the same argument,

N_{N}(P)/Q = C_{N_{N} (Q)/Q}(P) .

Since [N_{N}(P), K] = 1, then C_{N_{N} (Q)/Q}(P) = C_{N_{N} (Q)/Q}(P K). Now, we have that N_{N}(Q)/Q is KP -isomorphic to the direct product of N_{S_{i}}(Q_{i})/Q_{i} and that these factors are transitively permuted by P. By Lemma 2.7, we know that

C_{N_{N} (Q)/Q}(KP) = C_{N_{N} (Q)/Q}(P)

if and only if

C_{N_{S}(Q_{1})/Q_{1}}(KP_{0}) = C_{N_{S}(Q_{1})/Q_{1}}(P_{0}) = N_{S}(P_{0})/Q_{1} .

We conclude that K acts trivially on N_{S}(P_{0})/Q_{1}. Since [Q_{1}, K] = 1, by coprime action we have that [K, N_{S}(P_{0})] = 1, as desired. Hence, we can apply the inductive hypothesis in G_{0}. Now, by using the notation of central products used in Section 5.1 of [8], we can write θ = θ_{1} ·. . .·θ_{t}, where θ_{i} ∈ Irr(S_{i}). Since θ is P -invariant, then we conclude that θ_{1} is P_{0}-invariant. Since the determinantal order and the degree of θ_{1} are coprime to p, we see that θ_{1} extends to a p′-degree character of SP_{0}. By induction, this character extends to G_{0}, and we conclude that θ_{1} is K-invariant. The same argument applies to every θ_{i}, and we conclude that θ is K-invariant. This finishes the proof of the theorem. □

The proof of Corollary B now is immediate:

Proof of Corollary B. Let Γ = GA be the semidirect product. Then Γ has a Hall π-subgroup G and a π-complement A. By coprime action, G has an A-invariant Sylow p-subgroup P, and all of them are C_{G}(A)-conjugate. Also N_{Γ}(P) = N_{G}(P)A and N_{Γ}(A) = C_{G}(A) × A. Suppose that [N_{G}(P), A] = 1.

Then N_{Γ}(P) ⊆ N_{Γ}(A) and by Theorem A, we have that all irreducible p′- degree characters of G extend to Γ and are, therefore, A-invariant. Conversely, if all p′-degree irreducible characters of G are A-invariant, then all of them extend to Γ since (|Γ : G|, |G|) = 1. Hence, by Theorem A, there is a Sylow p-subgroup P_{1} of G such that N_{Γ}(P_{1}) ⊆ N_{Γ}(A). Hence [N_{G}(P_{1}), A] = 1. In particular, P_{1} is A-invariant, and we conclude that (P_{1})^{c} = P for some c ∈ C_{G}(A). Then [N_{G}(P), A] = 1, and the proof is complete. □

3. Proof of Theorem 2.2

The aim of this section is the proof of Theorem 2.2 which we restate as follows:

Theorem 3.1. Let S be quasi-simple, normal in a group X with S/Z(S) ≤ X/Z(S) ≤ Aut(S/Z(S)), and let p be a prime dividing |S|, with |Z(S)| prime to p. Let M/S be normal in X/S of order prime to |S|, such that X/M is a p-group. Let P ∈ Syl_{p}(X). Then the following are equivalent:

(i) all P -invariant characters in Irr_{p′}(S) are M-invariant; (ii) there exists a complement K of S in M normalized by P with

[N_{S}(P), K] = 1.

The statement is trivially true when M = S, so we may assume that the quasi-simple group S has outer automorphisms of order prime to |S|, which by the classification forces S to be of Lie type and M/S to consist of field automorphisms:

Proposition 3.2. Let S be finite quasi-simple and σ ∈ Aut(S) with gcd(o(σ), |S|) = 1. Then S is of Lie type and σ is a field automorphism. In particular, Out(S) has a cyclic central π′-Hall subgroup, where π = π(|S|).

Proof. According to [4, Cor. 5.1.4], any automorphism of a finite quasi-simple group S is induced by an automorphism of the simple quotient S/Z(S), so it suffices to deal with the case that S is simple. Since the outer automorphism group of finite simple groups not of Lie type has order a power of 2, clearly S must be of Lie type. In this case, Out(S) is described in [4, §2.5]: it is an extension of the normal subgroup D of diagonal automorphisms (which is cyclic or a Klein four group) by the commuting product of the cyclic group of field automorphisms with the group of graph automorphisms (the latter being a subgroup of 𝔖_{3}). Now the order of D is only divisible by prime divisors of the order of the Weyl group of S. But by Remark 3.3 below, since o(σ) is prime to |S|, it is only divisible by primes larger than those occurring in |D|, so σ can only act trivially on D. This shows the claim. □

Remark 3.3. In the preceding proof, we used the following observation: if W is a Weyl group, and p a prime divisor of its order, then all primes dividing p − 1 also divide |W |. This can easily be checked by inspection. It also follows from the rationality of Weyl groups. In fact, even more is true: all primes smaller than p divide |W |; no a priori proof of this is known to the authors.

3.1. The general setup. We fix the following notation throughout this section. Let G be a simple linear algebraic group of simply connected type over the algebraic closure of a finite field of characteristic r, and T ≤ G a maximal torus. Then for any graph automorphism of the Dynkin diagram of G and any integral power q = r^{a} there exists a Steinberg endomorphism F_{a} : G → G such that F_{a} acts as qϕ on the character group X(T) of T, with ϕ of finite order inducing the given graph automorphism on the Weyl group. Similarly, if G is of type B_{2}, G_{2} or F_{4}, and r = 2, 3, respectively 2, then for any odd power q = \sqrt{r}^{a} of \sqrt{r} there exists a Steinberg endomorphism F_{a} : G → G such that F_{a} acts as qϕ on X(T)⊗_{ℤ} ℝ and ϕ induces the non-trivial graph automorphism of the Coxeter diagram of G (see [15, Thm. 22.5], for example). Now for a Steinberg endomorphism F = F_{a} as above, let G = G^{F} be the corresponding finite group of fixed points. Then, with a finite number of exceptions, G is a quasi-simple group of Lie type, and all finite simple groups of Lie type arise by this setup as G/Z(G), except for ^{2}F_{4}(2)′. Since the latter group has no coprime automorphisms, this exception is of no importance for our question. So we may and will now assume that G = G^{F} is perfect. Then, except for a finite number of cases, G is the universal Schur cover of the simple group G/Z(G). Since none of the simple groups of Lie type with exceptional Schur multiplier have coprime automorphisms (see [4, §6.1]), we may assume that G/Z(G) is none of these. Thus, the quasi-simple group S from the statement of Theorem 3.1 can be realized as a central quotient S = G/Z of a group G as above, for some Z ≤ Z(G), and in particular all irreducible characters of S occur among the characters of G. According to Proposition 3.2 we will have to consider field automorphisms of S. Now any coprime field automorphism γ of G, and hence of S, is induced by a Steinberg endomorphism of G as follows. For fixed r and G, we may assume the F_{a} to be chosen compatibly such that F_{an} = F_{a}^{n} for any n coprime to o(ϕ). In this setting, if γ is a coprime field automorphism of G of order f, then in particular, f is prime to the order of ϕ, and thus there exists a Steinberg endomorphism F_{c} : G → G such that F = F_{cf} = F_{c}^{f} and ⟨γ⟩ = ⟨F_{c}|_{G}⟩ in Out(G). The centralizer of γ in G is then just the fixed point group G^{F_{c}} under F_{c}, a finite group of Lie type of the same type as G.

3.2. Action on Irr(G). In order to determine the action of coprime automorphisms on Irr(G), we first recall Lusztig’s parametrization. For this, let G^{∗} be a group in duality with G, with compatible Steinberg endomorphisms F_{a}^{∗} : G^{∗} → G^{∗}, a ≥ 1 (respectively a odd, when G is a Suzuki or Ree group). We’ll write G^{∗} := G^{∗}^{F^{∗}} for the group of fixed points of F^{∗} = F_{a}^{∗}. Then by the results of Lusztig, there is a partition

Irr(G) = ⨆_{s∈G^{∗}/∼} E(G, s)

into Lusztig series E(G, s), indexed by semisimple elements s of the dual group G^{∗} modulo conjugation. Now recall that S = G/Z for some subgroup Z ≤ Z(G). Let T ≤ G be a maximal torus, with dual torus T^{∗} ≤ G^{∗}. So any s ∈ T^{∗} is a linear character of T, and χ ∈ E(G, s) is trivial on Z ≤ T if and only if Z ≤ ker(s) (see [13, Lemma 2.2]). This defines a subgroup of T^{∗} of index |Z|; we denote by S^{∗} the normal subgroup of G^{∗} generated by all these subgroups, so G^{∗}/S^{∗} ≅ Z (and hence [G^{∗}, G^{∗}] ≤ S^{∗}). Thus, for s ∈ G^{∗} semisimple, the characters in E(G, s) have Z in their center, so descend to S = G/Z, if and only if s ∈ S^{∗}.

\xymatrix{G_{ss}^{∗} \ar@{.>}[r]^{param. irred. of} & G \ar@{->>}[d] \\ S_{ss}^{∗} \ar@{^{(}->}[u] \ar@{.>}[r]^{param. irred. of} & S}

Now any field automorphism of G permutes the various Lusztig series E(G, s). More precisely, if γ is induced by F_{c} as described above, then by Lusztig E(G, s) is γ-stable if and only if the G^{∗}-class of s is F_{c}^{∗}-stable. We need the following:

Lemma 3.4. In the above situation, assume that E(G, s) is γ-stable. Then γ fixes each orbit in E(G, s) under diagonal automorphisms.

Proof. Let H = C_{G^{∗}}(s). Lusztig [12, Prop. 5.1] proves the existence of a surjection

ψ : E(G, s) → E(H^{∘F} , 1) (mod H^{F} /H^{∘F})

with fibres the orbits under the action of diagonal automorphisms, such that multiplicities of ρ ∈ E(G, s) in Deligne–Lusztig characters are determined by those of elements of ψ(ρ). We claim that ψ can be chosen γ-equivariant. The result then follows since field automorphisms act trivially on E(H^{∘F} , 1) (see e.g. [2, Prop. 6.6]). Now unipotent characters are uniquely determined by their multiplicities in Deligne–Lusztig characters, and thus the claim follows, unless H has a component of exceptional type (see [2, Prop. 6.3]). But in the latter case, characters with same multiplicities in all Deligne–Lusztig characters have distinct eigenvalues of Frobenius attached, and since γ commutes with F, these are respected by γ. □

In our situation, as remarked in the proof of Proposition 3.2, any prime divisor of o(γ) is larger than the order of the group of diagonal automorphisms. Thus γ must in fact fix all elements in E(G, s). We hence get the following characterization of γ-stable characters of S (recall that all irreducible characters of S occur among those of G):

Proposition 3.5. Let γ be a coprime automorphism of S, where S = G/Z with G = G^{F}, Z ≤ Z(G) as above. Then χ ∈ Irr(S) is γ-stable if and only if χ ∈ E(G, s) for some F_{c}^{∗}-stable semisimple element s ∈ S^{∗}, where γ is induced by F_{c}.

Proof. This follows from the above, using the fact that any F_{c}^{∗}-stable conjugacy class of the connected group G contains an F_{c}^{∗}-stable (and hence F^{∗} = F_{c}^{∗}^{f}- stable) element (see e.g., [15, Thm. 21.11]). □

3.3. Defining characteristic. We now first dispose of an easy case; observe that in defining characteristic, all semisimple characters lie in Irr_{p′}(G), by Lusztig’s Jordan decomposition of characters (but note that when r is bad Irr_{p′}(G) may contain non-semisimple characters).

Proposition 3.6. Let S be quasi-simple of Lie type, γ a coprime automorphism of S. Let p = r, the defining characteristic of S. Let A be a p-group of automorphisms of S. Then γ does not centralize a Sylow p-subgroup of S, and moreover not all A-invariant elements of Irr_{p′}(G) are γ-invariant. In particular, Theorem 3.1 holds in this situation.

Proof. Let f denote the order of γ. As above, we may choose F, F_{c} : G → G such that S is a central quotient of G = G^{F}, F = F_{c}^{f} and γ is induced by F_{c}|_{G}. Then G_{0} := G^{F_{c}} is a group of the same type as G, but over a subfield 𝔽_{q_{0}} of 𝔽_{q}, of index [𝔽_{q} : 𝔽_{q_{0}}] = f. The order formula for groups of Lie type (see e.g., [15, Table 24.1]) then shows that the p-parts of G and G_{0} differ. Since Z(G) has order prime to p, the same holds for the p-parts of S and C_{S}(γ). In particular, γ does not centralize a Sylow p-subgroup of S, whence (ii) of Theorem 3.1 does not hold. Since diagonal automorphisms have order prime to the defining characteristic, A consists of graph and field automorphisms only. Thus, A induces a group of automorphisms A^{∗} on the dual G^{∗} of G, and we write H := C_{G^{∗}}(A^{∗}) for its fixed point group. Now let s ∈ H ∩ [G^{∗}, G^{∗}] be semisimple, not centralized by γ^{∗} (which exists by Zsigmondy’s theorem). Then by Proposition 3.5 the Lusztig series E(G, s) consists of characters with Z(G) in their kernel, invariant under A, but not invariant under γ. Moreover, the semisimple character in E(G, s) is of p′-degree. This shows that (i) of Theorem 3.1 is not satisfied either, thus completing the proof. □

3.4. Sylow subgroups. We now turn to the non-defining primes. There we require the following crucial result from [13, Prop. 7.3]:

Proposition 3.7. Let G be as above, p ≠ r and χ ∈ Irr_{p′}(G). Then there is some semisimple s ∈ G^{∗} centralizing a Sylow p-subgroup of G^{∗} with χ ∈ E(G, s). Conversely, if s ∈ G^{∗} centralizes a Sylow p-subgroup, then there is some χ ∈ E(G, s) of p-height 0.

We next consider the following generic case, which occurs for all large enough primes p:

Proposition 3.8. Let S = G/Z be quasi-simple of Lie type, with G = G^{F}, Z ≤ Z(G) as above. Let p ≠ r and P a Sylow p-subgroup of G, P^{∗} a Sylow p-subgroup of G^{∗}.

(a) Then P is contained in a proper F -stable Levi subgroup of G if and only if P^{∗} is contained in a proper F^{∗}-stable Levi subgroup of G^{∗}.

(b) Assume that the condition in (a) is satisfied, and that moreover S has no diagonal or graph automorphism of order p. Then neither (i) nor (ii) in Theorem 3.1 holds.

Proof. Let L be a proper F -stable Levi subgroup of G containing P. If L^{∗} denotes an F^{∗}-stable Levi subgroup of G^{∗} dual to L, then |G^{∗}| = |G| and |L^{∗}^{F^{∗}}| = |L^{F} |, so L^{∗} contains a Sylow p-subgroup P^{∗} of G^{∗}. Since we may exchange the roles of G, G^{∗} in this argument, we obtain (a). Now assume that P is contained in the F -stable proper Levi subgroup L. Then T := Z(L) is an F -stable torus of dimension at least 1 (see [15, Prop. 12.6]). Thus, for a coprime automorphism γ of S, induced by F_{c}, T^{F} \ T^{F_{c}} is nonempty. Let κ : G → G/Z = S denote the canonical epimorphism. Since κ(T^{F}) ≤ C_{S}(κ(P)) ≤ N_{S}(κ(P)), this shows that N_{S}(κ(P)) is not centralized by γ. On the other hand, the non-trivial torus T^{∗} := Z(L^{∗}) lies in C_{G^{∗}}(P^{∗}). It follows that there exists a semisimple s ∈ T^{∗}^{F^{∗}} \T^{∗}^{F_{c}^{∗}} ⊆ C_{G^{∗}}(P^{∗}) which hence is not F_{c}^{∗}-invariant. Then, by Proposition 3.5, the elements of E(G, s) are not γ-invariant. But E(G, s) contains characters of height 0 by Proposition 3.7 above. Furthermore, by assumption the Sylow p-subgroup B of Out(S) consists of field automorphisms, hence is cyclic. Thus there exists F_{b} : G → G such that β := F_{b}|_{G} generates B modulo inner automorphisms. Then we may argue as before with semisimple elements in T^{F_{b}} \ T^{⟨F_{b},F_{c}⟩}, respectively s ∈ T^{∗}^{F_{b}^{∗}} \ T^{∗}^{⟨F_{b}^{∗},F_{c}^{∗}⟩} to see that neither (i) nor (ii) of Theorem 3.1 are satisfied. □

3.5. Bad primes and torsion primes. It remains to deal with the primes p not satisfying the condition in Proposition 3.8(a). As announced before, these are small:

Proposition 3.9. Let H be a simple algebraic group in characteristic r with Steinberg endomorphism F : H → H. Let p ≠ r be a prime and P a Sylow p-subgroup of H := H^{F}. If P is not contained in any proper F -stable Levi subgroup of H, then every semisimple element centralizing P is quasi-isolated, and in particular, p is a bad prime or torsion prime for H.

Proof. Clearly we have that P ≠ 1. Let g ∈ C_{H}(P) be semisimple, so P ≤ C_{H}(g). Now the connected group C_{H}^{∘}(g) acts by conjugation on the set Ω of proper Levi subgroups of H containing C_{H}(g), and any orbit therein is F -stable; hence by [15, Thm. 21.11], if Ω is non-empty then it contains an F - stable element L, and then P ≤ C_{H}(g) ≤ L, which is not the case. So C_{H}(g) is not contained in any proper Levi subgroup of H, whence, by definition, g is quasi-isolated in H (see [15, Exmp. 14.4(2)]). Apply this to 1 ≠ g ∈ Z(P). If p is not a torsion prime for H, the index |C_{H}(g) : C_{H}^{∘}(g)| is prime to p (see [15, Prop. 14.20]), so P ≤ C_{H}^{∘}(g). By assumption, this does not lie in any proper Levi subgroup of H, so by definition g is isolated. Now by a result of Deriziotis (see [15, Rem. 14.5]) and the algorithm of Borel and de Siebenthal (see [15, Thm. 13.12]), this implies that all prime divisors of o(g) are bad for H. □

We need the following two elementary results, where we write Φ_{m} for the mth cyclotomic polynomial:

Lemma 3.10. Let q ≥ 1, p a prime not dividing q, and e the multiplicative order of q modulo p. Then p|Φ_{d}(q) if and only if d = ep^{i} for some i ≥ 0.

See e.g. [13, Lemma 5.2] for a proof.

Lemma 3.11. Let m, f ≥ 1. Then

Φ_{m}(X^{f}) = ∏_{d} Φ_{d}(X)

where the product runs over all divisors d of mf which do not divide m′f for any m′ < m. In particular, when gcd(m, f) = 1 then Φ_{m}(X^{f}) = ∏_{d|f} Φ_{md}(X).

Proof. The roots of Φ_{m}(X) are the primitive mth roots of unity. Thus the roots of Φ_{m}(X^{f}) are mfth roots of unity whose fth power is a primitive mth root of unity, hence whose order is not a divisor of m′f, for any m′ < m. □

Proposition 3.12. Let G be connected reductive with Steinberg endomorphisms F_{c}, F = F_{c}^{f} such that gcd(f, |G^{F} |) = 1, H ≤ G a connected reductive F_{c}-stable subgroup, p a prime dividing the order of the Weyl group of G, p ≠ r. Then H^{F_{c}} contains a Sylow p-subgroup of H^{F}.

Proof. The group H_{0} = H^{F_{c}} is a group of the same type as H = H^{F}. Thus there are non-negative integers a(d) such that |H_{0}|_{r′} = ∏_{d} Φ_{d}(q_{0})^{a(d)} and |H|_{r′} = ∏_{d} Φ_{d}(q)^{a(d)}, with q_{0}^{f} = q, and d divides the order of the Weyl group W of G whenever a(d) > 0 (see [15, Cor. 24.6 and 24.7]). Since f is coprime to |H|, it’s coprime to p − 1 by Remark 3.3, so q, q_{0} have the same order e modulo p. Now, if p divides Φ_{d}(q) = Φ_{d}(q_{0}^{f}), then d = ep^{i} for some i ≥ 0 by Lemma 3.10, and hence gcd(f, d) = 1. Then the only factor of Φ_{d}(q_{0}^{f}) (in the factorization given by Lemma 3.11), which is divisible by p, is Φ_{d}(q_{0}) (again using Lemma 3.10). Thus the p-parts of Φ_{d}(q) and Φ_{d}(q_{0}) coincide, and hence |H_{0}|_{p} = |H|_{p}. □

Corollary 3.13. Let S be quasi-simple of Lie type as above, γ a coprime automorphism and p a prime dividing the order of the Weyl group of S. Then the fixed group C_{S}(γ) contains a Sylow p-subgroup of S.

Proof. Write S = G/Z with G = G^{F}, Z ≤ Z(G). Now γ is induced by some Steinberg endomorphism F_{c} with F_{c}^{f} = F, where f = o(γ) is coprime to |S|. Then |G_{0}|_{p} = |G|_{p} by the preceding result, with G_{0} = G^{F_{c}}. Since the subgroup Z ≤ Z(G) has order only divisible by prime divisors of the order of the Weyl group, it is fixed by F_{c} (again by Remark 3.3) and hence G_{0}/(Z∩G_{0}) = G_{0}/Z ≤ C_{S}(γ), which shows that |S|_{p} = |C_{S}(γ)|_{p}. □

After these preparations we can return to the proof of Theorem 3.1.

Proposition 3.14. Let S = G/Z be quasi-simple of Lie type, with G = G^{F} and Z ≤ Z(G) as above, p ≠ r a prime such that no proper F -stable Levi subgroup of G contains a Sylow p-subgroup of G. Then (i) in Theorem 3.1 holds.

Proof. Let χ ∈ Irr_{p′}(S), so χ ∈ E(G, s) for some semisimple element s ∈ G^{∗} centralizing a Sylow p-subgroup P^{∗} of S^{∗}, by Proposition 3.7. Now by Proposition 3.8, P^{∗} is not contained in a proper F^{∗}-stable Levi subgroup of G^{∗}, so by Proposition 3.9, applied to H = G^{∗}, we conclude that all semisimple elements in C_{G^{∗}}(P^{∗}) are quasi-isolated, their order is only divisible by torsion primes and bad primes, so in particular only by divisors of the order of the Weyl group of G^{∗}. Let T^{∗} be an F^{∗}-stable torus of G^{∗} containing s. Then Proposition 3.12 applied with H = T^{∗} shows that s ∈ T^{∗}^{F_{c}^{∗}}. But then χ is F_{c}-stable by Proposition 3.5. □

For p odd let e_{p}(q) denote the order of q modulo p, respectively the order of q modulo 4 when p = 2.

Proposition 3.15. Let S = G/Z be quasi-simple of Lie type, with G = G^{F} and Z ≤ Z(G) as above, p ≠ r a prime such that no proper F -stable Levi subgroup of G contains a Sylow p-subgroup of G. Then (ii) in Theorem 3.1 holds.

Proof. By Proposition 3.9 our assumptions imply that p is a torsion or bad prime for S. By [13, Thms. 5.14, 5.19, 8.4] the normalizer of a Sylow p-subgroup S_{p} of S is contained in the normalizer of a Φ_{e}-torus T_{e}, where e = e_{p}(q), unless S = SL_{3}(q), SU_{3}(q), G_{2}(q),^{2}F_{4}(q^{2}) with p = 3 or S = Sp_{2n}(q),^{2}G_{2}(q^{2}) with p = 2. Now first assume that S is of exceptional type. Since a Sylow 2-subgroup of (2)E_{6}(q) (with q odd) is contained in a Levi subgroup of type ^{(2)}D_{5}(q), and a Sylow 3-subgroup of E_{7}(q) (with 3 |̸q) is contained in a Levi subgroup of type E_{6}(q) or ^{2}E_{6}(q), these situations do not arise here. In the remaining cases, which are collected in Table 1, the Sylow normalizers may easily be worked out explicitly inside N_{S}(T_{e}), respectively they are already given in [14, Sect. 3 and 4] (see also [11, Sect. 4] for the case p = 2). In particular they depend only on the integer a, which is the same for S and for the centralizer of any coprime automorphism by Corollary 3.13. Now consider the case where S is of classical type. If S is not of type A, then p = 2 is the only torsion prime, and from the above mentioned result on N_{S}(S_{p}) (or from [1, Thm. 4]) it follows that a Sylow 2-subgroup S_{2} of S is self-normalizing, or an extension of S_{2} of degree 3^{t} where t is the number of summands in the 2-adic expansion of n for S = Sp_{2n}(q) with q ≡ ±3 (mod 8). In particular, it’s the same in S and in C_{S}(γ). Finally, when S is linear the possible torsion primes are divisors of gcd(n, q − 1). For SL_{3}(q) with p = 3 and q ≡ 4, 7 (mod 9) the Sylow p-normalizers is isomorphic to 3^{1+2}.Q_{8}, independently of q (see [14, Sect. 3.1]), whence the Table 1. Normalizers for Sylow subgroups of exceptional groups not contained in proper Levi subgroups

G  p N(S_{p})
2G_{2}(3^{2f+1}) 2 2^{3}.7.3
G_{2}(q)  2 (2^{a})^{2}.2^{2}
3 \begin{cases}(3^{a})^{2}.W (G_{2}) & q ≡ 1, 8 (mod 9) \\ 3^{1+2}.Q_{8}.2 & q ≡ 2, 4, 5, 7 (mod 9) \end{cases}
3D_{4}(q)  2 (2^{a})^{2}.2^{2}
3 (3^{a} × 3^{a+1}).W (G_{2})
2F_{4}(2^{2f+1}) 3 \begin{cases}(3^{a})^{2}.W (G_{2}) & 2^{2f+1} ≡ 8 (mod 9) \\ 3^{1+2}.Q_{8}.2 & 2^{2f+1} ≡ 2, 5 (mod 9) \end{cases}
F_{4}(q)  2 (2^{a})^{4}.S_{2}(W (F_{4}))
3 (3^{a})^{4}.3^{2}.2^{3}
E_{6}(q)  3 \begin{cases}(3^{a})^{6}.S_{3}(W (E_{6})).2^{2} & q ≡ 1 (mod 3) \\ (3^{a})^{4}.3^{2}.2^{3} & q ≡ 2 (mod 3) \end{cases}

2E_{6}(q)  3 \begin{cases}(3^{a})^{6}.S_{3}(W (E_{6})).2^{2} & q ≡ 2 (mod 3) \\ (3^{a})^{4}.3^{2}.2^{3} & q ≡ 1 (mod 3) \end{cases}
E_{7}(q)  2 (2^{a})^{7}.S_{2}(W (E_{7}))
E_{8}(q)  2 (2^{a})^{8}.S_{2}(W (E_{8}))
3 (3^{a})^{8}.S_{3}(W (E_{8})).2^{4}
5 \begin{cases}(5^{a})^{8}.S_{5}(W (E_{8})).[2^{6}] & q ≡ ±1 (mod 5) \\ (5^{a})^{4}.5.4^{2} & q ≡ ±2 (mod 5) \end{cases}

Here, p^{a} = |Φ_{e}(q)|_{p} with e = e_{p}(q), respectively 3^{a} = |Φ_{4}(2^{2f+1})|_{3} for the Ree groups, and

S_{p}(H)

denotes a Sylow p-subgroup of H. Moreover, p^{a} stands for a cyclic group of that order, [p^{k}]

for an un-

specified group of order p^{k}, H^{k} for a direct product of k copies of H.

claim holds. Exclude this case. Let n = ∑_{i} p^{k_{i}} be the p-adic expansion of n (thus, any p^{k_{i}} occurs at most p − 1 times). If this has at least two summands, then a proper Levi subgroup GL_{n_{1}}(q) ∘ GL_{n_{2}}(q), with n_{1} = p^{k_{1}}, n_{2} = n − n_{1}, contains a Sylow p-subgroup of S. Thus we may assume that n = p^{k} is a p-power. For p = 2 the Sylow 2-subgroups are self-normalizing by [1, Thm. 4]. For odd p an easy matrix calculation shows that the normalizer of S_{p} (modulo the center) is an extension of a homocyclic group (p^{a})^{n−1} by the normalizer in 𝔖_{n} of one of its Sylow p-subgroups, where p^{a} = |q − 1|_{p}. So again, it is the same in S as in the centralizer of any coprime automorphism. The case of unitary groups is entirely similar. □

The proof of Theorem 3.1 is now complete by Propositions 3.6, 3.8, 3.14 and 3.15. In the course of the proof we have established the following: if S is of Lie type and p not the defining characteristic, then (i) and (ii) in Theorem 3.1 are equivalent to

(iii) a Sylow p-subgroup of S is not contained in any proper Levi subgroup

of S.

Acknowledgement. We wish to thank the anonymous referee for a very careful reading of a previous version and for proposing several improvements.

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Gunter Malle FB Mathematik TU Kaiserslautern Postfach 3049 67653 Kaiserslautern Germany. malle@mathematik.uni-kl.de Gabriel Navarro Departament d’Àlgebra Universitat de València Dr. Moliner 50 46100 Burjassot Spain. gabriel.navarro@uv.es 920

Documenta Mathematica 16 (2011)