REAL CHARACTERS OF p′-DEGREE

by

I. M. Isaacs Department of Mathematics

University of Wisconsin

480 Lincoln Dr Madison, WI 43706 USA E-mail: isaacs@math.wisc.edu

Gunter Malle

Fachbereich Mathematik/Informatik Universität Kassel Heinrich-Plett-Str. 40

D-34132 Kassel GERMANY E-mail: malle@mathematik.uni-kassel.de

Gabriel Navarro Departament d’Àlgebra Facultat de Matemàtiques Universitat de València

46100 Burjassot. València SPAIN E-mail: gabriel@uv.es

The research of the first author was partially supported by the U. S. National Security Agency and that of the third author by the Ministerio de Ciencia, grant BFM2001-1667-C03-02

1. Introduction.

A classical theorem of W. Burnside asserts that a finite group has a nonprincipal real-valued irreducible character if and only if the group has even order. In this note we seek a p-local analog of this result, where p is prime. We ask, in other words, when it is true that a finite group has a nonprincipal real-valued irreducible character of p′-degree, and we settle this completely except when p = 3.

THEOREM A. Let G be a finite group and suppose that p ≠ 3 is prime. If G is nonsolvable, then it has a nonprincipal, real, irreducible character of p′-degree. If G is solvable, then it has such a character if and only if |N_{G}(P)| is even, where P ∈ Syl_{p}(G).

The first conclusion of the Theorem A is actually false when p = 3. A counterexample is the group G formed by taking the semidirect product of L_{2}(27) acted on by the cyclic group generated by a field automorphism of order 3. Although G is nonsolvable, every nonprincipal, real irreducible character has degree divisible by 3. (To check this, it suffices to observe that all nonprincipal real characters of L_{2}(27) that are fixed by the field automorphism of order 3 have degree divisible by 3.)

The prime p = 3 is not an exception to the second assertion of Theorem A, and furthermore, it is not necessary to assume that G is solvable; p-solvability suffices. The following is thus a strong form of the second part of Theorem A.

THEOREM B. Suppose that G is a finite p-solvable group, where p is prime. Then G has a nonprincipal, real-valued, irreducible character of p′-degree if and only if |N_{G}(P)| is even, where P ∈ Syl_{p}(G).

As might be expected, the proof of the first part of Theorem A reduces to a question about simple groups. It is easy to dispose of the alternating groups and the sporadic groups, and so by the classification, it suffices to study groups of Lie type. This is done in the final section of this paper. Our proof of Theorem B also relies on the simple group classification, but only indirectly; it quotes a previously known result.

2. Proving that Sylow normalizers have even order.

The main result of this section is Theorem 2.2, which proves the half of Theorem B that asserts that if G is p-solvable and has a nonprincipal, real, irreducible character of p′-degree, then a Sylow p-normalizer in G must have even order. This, of course, also establishes part of Theorem A.

We will need the following consequence of the Glauberman correspondence. (See Chapter 13 of [4].)

(2.1) LEMMA. Let A be solvable and act via automorphisms on G, where A and G have coprime orders. Then A fixes a nonprincipal, real, irreducible character of G if and only if |C_{G}(A)| is even.

Proof. Write C = C_{G}(A) and recall that the Glauberman correspondence is a canonical bijection ψ ↦ ψ∗, from the set of A-invariant irreducible characters of G onto Irr(C). Also, (1_{G})^{∗} = 1_{C} since in general, ψ^{∗} is an irreducible constituent of ψ_{C}. Because the Glauberman correspondence is canonically defined, it respects field automorphisms, and we conclude that ψ is real if and only if ψ^{∗} is real. Thus A fixes a nonprincipal, real, irreducible character of G if and only if C has a nonprincipal, real, irreducible character. By Burnside’s result, however, C has such a character if and only if |C| is even.

The key step in the proof of Lemma 2.1 is that because the Glauberman map is canonical, (whatever that means, exactly) it respects field automorphisms. Unfortunately, the Glauberman map is somewhat difficult to describe in general, and it is not quite a triviality to show that it is well defined. In this paper, however, we use Lemma 2.1 only when A is a p-group, and in that special case, it is easy to describe the map and to see that it respects field automorphisms: ψ^{∗} is the unique irreducible constituent of ψ_{C} that occurs with multiplicity not divisible by p. The case of Lemma 2.1 that we actually need, therefore, is more elementary than the general result stated above.

(2.2) THEOREM. Let G be p-solvable, where p is prime, and let P ∈ Syl_{p}(G). Suppose that G has a nonprincipal, real, irreducible character with p′-degree. Then N_{G}(P) has even order.

Proof. We proceed by induction on |G|. Let N be a minimal normal subgroup of G and note that P N/N ∈ Syl_{p}(G/N). Also, it follows via the Frattini argument that N_{G/N}(P N/N) = N_{G}(P)N/N, and so if |N_{G/N}(P N/N)| is even, we see that |N_{G}(P)| is even, and there is nothing further to prove. We can assume, therefore, that |N_{G/N}(P N/N)| is odd, and thus by the inductive hypothesis, we see that the group G/N cannot have a nonprincipal, real, irreducible character of p′-degree.

Let χ ∈ Irr(G) be nonprincipal and real, and suppose that p does not divide χ(1). Since the group G/N has no such character, it follows that N ⊈ ker(χ), and thus we can choose a nonprincipal, irreducible constituent θ of χ_{N}. By the Clifford correspondence, χ is induced from G_{θ}, the stabilizer of θ in G, and hence |G : G_{θ}| divides χ(1), which is a p′-number. It follows that G_{θ} contains a full Sylow p-subgroup of G. It is thus no loss to assume that P ⊆ G_{θ}, and hence θ is P -invariant.

Consider the (nonempty) set S of P -invariant irreducible constituents of χ_{N}. If φ ∈ S, then since both θ and φ are irreducible constituents of χ_{N}, we can write φ = θ^{g} for some element g ∈ G. Now P stabilizes θ, and thus P^{g} stabilizes θ^{g} = φ. But φ ∈ S, and thus P also stabilizes φ, and we see that P, P^{g} ∈ Syl_{p}(G_{φ}). By Sylow’s theorem, therefore, we can write (P^{g})^{x} = P, for some element x ∈ G_{φ}. Then gx ∈ N_{G}(P), and we see that θ^{gx} = φ^{x} = φ. Since φ ∈ S was arbitrary, it follows that N_{G}(P) acts transitively on S. If |S| is even, therefore, then |N_{G}(P)| is even, and there is nothing more to prove.

We can now assume that |S| is odd. Because χ is real valued, it follows that the set S of P -invariant irreducible constituents of χ_{N} is permuted by complex conjugation. Since |S| is odd, S must contain a real character, and so it is no loss to assume that θ is real.

We are assuming that G is p-solvable, and so either N is a p-group, or else it is a p′-group. If N is a p-group, then since θ is a nonprincipal, real, irreducible character of N, it follows by Burnside’s result that |N| is even (and hence p = 2). But N_{G}(P) contains the p-group N, and so we see that |N_{G}(P)| is even, as wanted.

Finally, we can assume that N is a p′-group, and we write C = C_{N}(P). Since θ is P - invariant, real and nonprincipal, it follows by Lemma 2.1 that |C| is even. But C ⊆ N_{G}(P), and the result follows.

3. Construction of real irreducible characters.

We start work now toward proving those parts of Theorem A and Theorem B that assert the existence (under appropriate additional hypotheses) of a nonprincipal, real, irreducible character with p′-degree.

Our first result in this direction handles the case where p = 2.

(3.1) LEMMA. Suppose that |G| is even. Then G has a nonprincipal, real, irreducible character of odd degree.

Proof. Let P ∈ Syl_{2}(G). Since P is a nontrivial 2-group, it has a nonprincipal real-valued linear character λ. Then λ^{G} is real valued, and hence its nonreal irreducible constituents occur in complex-conjugate pairs. Since the degree λ^{G}(1) = |G : P | is odd, it follows that there exists a real irreducible constituent χ of λ^{G} that has odd degree. Finally, χ is not principal since it lies over the nonprincipal character λ ∈ Irr(P).

Since by the Feit-Thompson theorem, nonsolvable groups must have even order, we see that for p = 2, the first part of Theorem A is a consequence of Lemma 3.1. Also, the part of Theorem B that we have not already established is immediate from Lemma 3.1 when p = 2. Although Theorems A and B have now been proved when p = 2, we will, nevertheless, include the case p = 2 in some of our results, below.

If θ is a character of a group G, recall that det(θ) denotes the (uniquely determined) linear character of G given by g ↦ det(X (g)), where X is any representation affording χ. Also, o(θ) is the order of det(θ) in the group of linear characters of G.

The following is elementary and is surely well known.

(3.2) LEMMA. Suppose that θ ∈ Irr(G) is real. Then o(θ) divides 2.

Proof. Let R be a representation affording θ. Then the complex-conjugate representation \overline{R} affords θ̄ = θ. We conclude that R and \overline{R} are similar, and hence they have equal determinants. But clearly, these determinates are also complex-conjugate characters, and so the linear character det(θ) is real. The lemma now follows.

The next result constructs real characters in situations where we have some solvability.

(3.3) THEOREM. Let N ⊆ M ◁ G, where N ◁ G and M/N is solvable. Let P ∈ Syl_{p}(G), where p is prime, and suppose that θ ∈ Irr(N) is real and P -invariant and that it has p′- degree. Assume also that |N_{G}(P) : N_{G}(P) ∩ N| is odd. Then there exists ψ ∈ Irr(M), lying over θ, and such that ψ is real and P -invariant and has p′-degree.

Proof. Observe that it suffices to assume that M/N is a chief factor of G, and thus since we are assuming that this section is solvable, it must be a q-group for some prime q. We proceed now by induction on |G|.

Let T = G_{θ}, the stabilizer of θ in G, and note that P ⊆ T, and thus P ∈ Syl_{p}(T). Also, |N_{T} (P) : N_{T} (P) ∩ N| divides |N_{G}(P) : N_{G}(P) ∩ N|, which is odd. If T < G, it follows by the inductive hypothesis applied in the group T that there exists η ∈ Irr(T ∩ M) such that η is real and P -invariant, has p′-degree and lies over θ. By the Clifford correspondence in M, it follows that ψ = η^{M} is irreducible, and it clearly lies over θ and is P -invariant and real valued. Furthermore, ψ(1) = η(1)|M : M ∩ T |, and this is not divisible by p because M ∩ T ⊇ M ∩ P ∈ Syl_{p}(M). There is nothing further to prove in this case, and so we can assume that θ is G-invariant.

Next, observe that if p = 2, then P ⊆ N, and thus every character of M is P -invariant. In this situation, θ has odd degree and M/N has odd order, and thus θ^{M} is real and has odd degree. It follows that θ^{M} has a real-valued irreducible constituent of odd degree and we are done in this case. We can assume, therefore, that p ≠ 2.

By Lemma 3.2, we know that o(θ) divides 2. If q = p, therefore, then both the degree and the determinantal order of θ are coprime to |M : N|, and thus θ has a unique extension ψ ∈ Irr(M) such that o(ψ) = o(θ). (See, for example, Corollary 6.28 of [4].) It follows from the uniqueness that ψ is P -invariant and real valued. Also, ψ(1) = θ(1) is not divisible by p, and we are done in this case. We can therefore assume that q ≠ p.

Now θ is invariant in G and M/N is an abelian chief factor, and thus by Problem 6.12 of [4], we know that either there is a unique member ψ ∈ Irr(M) that lies over θ or else θ extends to M. In the first situation, it follows from the uniqueness that ψ is P -invariant and real valued. Also, since ψ(1)/θ(1) is a power of q, we see that ψ(1) is not divisible by p. There is nothing more to prove in this case, and so we can assume that θ has an extension θ̂ ∈ Irr(M).

By Gallagher’s theorem (Corollary 6.17 of [4]) every extension of θ to M is uniquely of the form λθ̂ for some linear character λ of M/N. Since M/N is abelian, the number of extensions of θ to M is equal to |M/N|, which is not divisible by p. It follows that θ has a P -invariant extension, and so we can assume that θ̂ is P -invariant.

The P -invariant extensions of θ to M are exactly the characters λθ̂, where λ is a P -invariant linear character of M/N. The number of P -invariant extensions of θ to M, therefore, is equal to the number of P -invariant linear characters of M/N. This number is |(M/N) : [(M/N), P]| = |C_{M/N}(P)|, which divides |N_{G/N}(P N/N)|. (The equality here follows from Fitting’s lemma since M/N is abelian and has order coprime to |P |.) By the Frattini argument, N_{G/N}(P N/N) = N_{G}(P)N/N, and since this group has odd order, we see that the number of P -invariant extensions of θ to M is odd.

Now θ is real, and thus the set of P -invariant extensions of θ to M is permuted by complex conjugation. We have just seen that this set has odd cardinality, and it follows that some P -invariant extension of θ to M is fixed by conjugation, and so is real, and this completes the proof.

To complete the proof of Theorem B, we need the following consequence of the classification of simple groups. (See the paper [7] of Y. Wang and Z. Chen.)

(3.4) THEOREM. Let P be a p-group that acts by automorphisms on a finite group G of order not divisible by p, and suppose that |C_{G}(P)| is odd. Then G is solvable.

The proof of this result proceeds by a fairly straightforward reduction to the case where G is simple. In that case, the argument depends on the fact that nontrivial coprime automorphisms of simple groups occur only for groups of Lie type over some field, and that all such automorphisms come from automorphisms of the field.

For Theorem B, we could, of course, avoid this appeal to the classification if we were willing to strengthen the hypothesis that G is p-solvable and assume instead that G is solvable. For the first part of Theorem A, however, the following deep result seems to be essential. (We will defer the proof to the next section.)

(3.5) THEOREM. Let P be a p-group, where p > 3, and suppose that P acts by automorphisms on a nonabelian simple group G. Then G has a nonprincipal, P -invariant, real, irreducible character of p′-degree.

The assertion corresponding to Theorem 3.5 is false when p = 3. A counterexample is given by the group G = L_{2}(27), where |P | = 3, acting by field automorphisms.

The following result includes the parts of Theorems A and B that we have not yet proved.

(3.6) THEOREM. Let P ∈ Syl_{p}(G), where p is prime, and assume either that p ≠ 3 or that G is p-solvable. Suppose that G fails to have a nonprincipal real irreducible character with p′-degree. Then |N_{G}(P)| is odd and G is solvable.

Proof. If p = 2, then by Lemma 3.1, we see that |G| is odd, and thus G is solvable and there is nothing further to prove. We assume, therefore, that p ≠ 2.

Let N be minimal normal in G. Then G/N fails to have a nonprincipal, real, irreducible character of p′-degree, and so if we work by induction on |G|, we can assume that G/N is solvable and that |N_{G/N}(P N/N)| is odd. Since N_{G/N}(P N/N) = N_{G}(P)N/N, it follows that |N_{G}(P) : N_{G}(P) ∩ N| is odd, and so the proof will be complete when we show that |N_{G}(P) ∩ N| is odd and that N is solvable. In particular, since p ≠ 2, we see that there is nothing further to prove in the case where N is a p-group, and so we can assume that N is not a p-group.

Since G has no nonprincipal, real, irreducible character of p′-degree, we can apply Theorem 3.3 to deduce that N has no nonprincipal, real, P -invariant, irreducible character of p′-degree. If p does not divide |N|, then all irreducible characters of N have p′-degree, and therefore no nonprincipal, real, irreducible character of N is P -invariant. By Lemma 2.1, therefore, |C_{N}(P)| is odd, and hence by Theorem 3.4, we see that N is solvable. In this case, N_{G}(P) ∩ N = C_{G}(P), and since this subgroup has odd order, we are done.

Finally, we can assume that the minimal normal subgroup N of G is neither a p-group nor a p′-group, and thus G is not p-solvable. By hypothesis, then, p ≠ 3, and so p > 3. We can write N = S_{1} × S_{2} × ⋯ × S_{r}, where the groups S_{i} are nonabelian simple and they are transitively permuted by P. Note that in this case, we want to derive a contradiction since we are trying to prove that N is solvable.

Let Q = N_{P} (S_{1}) and apply Theorem 3.5 to find a Q-invariant character φ_{1} ∈ Irr(S_{1}), where φ_{1} is nonprincipal, real and of p′-degree. Now for each subscript i with 1 ≤ i ≤ r, choose x_{i} ∈ P so that (S_{1})^{x_{i}} = S_{i}, and let φ_{i} = φ^{x_{i}} ∈ Irr(S_{i}). Observe that because φ_{1} is Q-invariant, it follows easily that the characters φ_{i} are unambiguously determined; they are independent of the choices of the elements x_{i} ∈ P.

We claim that the action of P permutes the characters φ_{i}. To prove this, let x ∈ P be arbitrary and suppose that (S_{i})^{x} = S_{j}. To see that (φ_{i})^{x} = φ_{j}, it suffices to check that (φ_{1})^{x_{i}x} = φ_{j}. This is true, however, since the element x_{i}x carries S_{1} to S_{j}, and so it is no loss to assume that x_{j} = x_{i}x. It follows from this that the product character θ = φ_{1} × ⋯ × φ_{r} is P -invariant.

Since φ_{1} is real, so are all of the φ_{i}, and it follows that θ is real, and similarly, since φ_{1} has p′-degree, it follows that θ has p′-degree. Also, of course, θ is nonprincipal, and this is our final contradiction.

4. Simple groups.

In this section we establish Theorem 3.5, which, we recall, concerns a p-group acting on a simple group, where p > 3. In fact, we prove a stronger and more general result.

(4.1) THEOREM. Let G be a finite simple group. Suppose that p > 2 is prime and write X to denote the set of nonprincipal, real, irreducible characters of G that have p′- degree. Then either there exists an Aut(G)-invariant character in X, or else p = 3 and G = L_{2}(3^{a}) for some integer a > 1. Furthermore, if in the exceptional case, a is not a power of 3, then it is possible to choose a member of X whose stabilizer in Aut(G) contains a Sylow 3-subgroup.

In particular, we see that except in the case G = L_{2}(3^{a}), where a is a power of 3, the conclusion of Theorem 3.5 remains valid even when p = 3. It is not hard to see that the exception is genuine. If p = 3 and a > 1 is a power of 3, the conclusion of Theorem 3.5 definitely fails when G = L_{2}(3^{a}) and P is the full group (of order a) of automorphisms of G induced by field automorphisms.

We proceed by checking the various possibilities for the simple group G (according to the classification). We start with the sporadic groups.

(4.2) LEMMA. Suppose that G is a sporadic finite simple group other than J_{1} and let D be the set of integers d > 1 such that there is exactly one irreducible character of G with degree d. Then gcd(D) = 1.

Proof. Check the Atlas [1].

An irreducible character of G that is unique of its degree is clearly rational valued and invariant under Aut(G). For each prime p and for each sporadic simple group G other than J_{1}, we see from Lemma 4.2 that G has a nonprincipal, irreducible character of p′- degree that is unique of its degree, and thus Theorem 4.1 is valid for all of these groups. If G = J_{1}, we check that it has a unique irreducible character that is real and of degree 76, and similarly for degree 77. By uniqueness, these two characters are Aut(G)-invariant, and since at least one of them has p′-degree for each prime p, the proof of Theorem 4.1 is complete in the case that G is sporadic.

Next, we handle the alternating groups.

(4.3) LEMMA. Let p > 2 be prime. If n ≥ 5, then the symmetric group S_{n} has a nonlinear irreducible character of p′-degree whose restriction to A_{n} is irreducible.

Proof. As is well known, the irreducible characters of S_{n} are parametrized by the partitions of n, and the characters that restrict irreducibly to A_{n} are exactly those corresponding to partitions whose graph (Young diagram) is not symmetric. Also, the degree of the irreducible character corresponding to a given partition can be computed using the standard hook-length formula.

Consider the partitions (n − 1, 1) and (n − 2, 2). Since n ≥ 5, neither of these is symmetric, and so each of the corresponding characters restricts irreducibly to A_{n}. The degrees of these characters are, respectively,

\frac{n!}{n(n − 2)!} = n − 1  and  \frac{n!}{2(n − 1)(n − 2)(n − 4)!} = \frac{n(n − 3)}{2} .

But since gcd(n − 1, n(n − 3)/2) is a power of 2 and p > 2, we see that at least one of these two characters has p′-degree.

Returning to Theorem 4.1 now, suppose that G = A_{n} with n ≥ 5. If n ≠ 6, then Aut(A_{n}) = S_{n}, and thus every irreducible character of A_{n} that is the restriction of a character of S_{n} is invariant under Aut(G). Also, of course, any such character is real (and in fact rational) since the characters of S_{n} are rational. It follows from Lemma 4.3, therefore, that Theorem 4.1 holds when G is any alternating group other than A_{6}. But A_{6} has a unique irreducible character of degree 9 and also one of degree 10, and so Theorem 4.1 holds for A_{6} too. (Actually, since A_{6} ≅ L_{2}(9), this group will also be covered below, when we consider groups of Lie type.)

There is one other simple group that requires individual attention: the Tits group 2F_{4}(2)′. According to the Atlas, this group has an irreducible character of degree 325 that establishes Theorem 4.1 for p = 3, and a character of degree 1728 that works for all other odd primes.

Finally, we deal with the remaining simple groups using Lusztig’s parametrization of the complex irreducible characters of finite reductive groups.

Proof of Theorem 4.1. We start by fixing some notation. Let G be a simple simplyconnected algebraic group over the algebraic closure of a finite field 𝔽_{q}, where q is a power of the prime ℓ, and let F : G → G be a Frobenius endomorphism with respect to some 𝔽_{q}-rational structure of G. Let Ĝ = G^{F} be the finite group of fixed points and assume that Ĝ/Z(Ĝ) is simple. By the classification, all nonabelian finite simple groups other than the alternating and sporadic groups and the Tits group occur in such a setting, and since we have already considered those groups, we can assume that G = Ĝ/Z(Ĝ) as above.

First assume that p ≠ ℓ and let χ be the Steinberg character of Ĝ. (See [2], Section 6, for example.) It follows from the defining formula for the Steinberg character that χ is real. Moreover, it is invariant under all automorphisms of G since these stabilize the set of parabolic subgroups. Also, ker(χ) = Z(Ĝ), and so χ defines a character of G. The degree of the Steinberg character χ is a power of ℓ, and hence it is prime to p, and we are done in this case.

It remains to consider the case ℓ = p > 2, and we note that the Suzuki groups and the Ree groups in characteristic 2 do not occur here since for these, ℓ = 2. Let G^{∗} be a group in duality with G, also defined over 𝔽_{q}, with corresponding Frobenius endomorphism F^{∗}, and let Ĝ^{∗} = (G^{∗})^{F^{∗}}. For every conjugacy class [s] of a semisimple element s ∈ Ĝ^{∗} there is a so-called semisimple (irreducible) character χ_{s} ∈ Irr(Ĝ) of degree prime to ℓ = p. (See 8.4.8 of [2] for the case where Z(Ĝ) is connected. A reference for the general case is [5].) (We mention that it follows from Lusztig’s parametrization of irreducible characters of Ĝ that these are essentially the only characters of p′-degree, up to very few exceptions when q = 2 or q = 3.)

If s is contained in the derived subgroup (Ĝ^{∗})′ of the dual group, then Z(Ĝ) ⊆ ker(χ_{s}), and so χ_{s} defines an irreducible character of G. Furthermore, by [5], this character is invariant under Aut(G) provided that the class [s] is invariant under Aut(Ĝ^{∗}) and the centralizer C_{G^{∗}}(s) is connected. Finally, χ_{s} is an explicit linear combination of Deligne-Lusztig generalized characters. (See 8.4.6 of [2].) It follows from this that χ_{s} is real if s is a real element; that is, if s is conjugate to its inverse. We are thus reduced to producing a real semisimple element in Ĝ^{∗}, lying in the derived subgroup, having connected centralizer in Ĝ^{∗} and such that its conjugacy class is invariant under Aut(Ĝ^{∗}).

Suppose first that Ĝ is one of the groups: G_{2}(q), ^{3}D_{4}(q), F_{4}(q), E_{6}(q), ^{2}E_{6}(q), E_{7}(q) or E_{8}(q). Let s be an involution with centralizer of type A_{1}(q)^{2}, A_{1}(q^{3})A_{1}(q), B_{4}(q), D_{5}(q), ^{2}D_{5}(q), D_{6}(q)A_{1}(q) or D_{8}(q), respectively. (See Table 4.5.1 of [3]). By [3], this involution has connected centralizer and lies in the derived group of Ĝ^{∗}. Moreover, by the classification of involutions in these groups, the class of s is invariant under the respective automorphism group in each case. Since involutions are real, the corresponding semisimple characters χ_{s} solve the problem.

We have already mentioned that we need not consider the Suzuki groups and the Ree groups in characteristic 2, and so the only remaing possibility among the exceptional groups is where G = Ĝ is the Ree group ^{2}G_{2}(q), where q is a power of 3. In this case, we can take for χ the semisimple character of degree q^{4} + q^{2} + 1, belonging to the class of involutions in Ĝ^{∗}. This is real and Aut(G)-invariant, and this completes the discussion of exceptional groups of Lie type.

We can now assume that G is a simple simply-connected group of classical type over a field of odd characteristic p, defined over the prime field 𝔽_{p} with corresponding Frobenius morphism F_{0} : G → G. The group Ĝ considered above is then the fixed group of some power F = F_{0}^{a} of this Frobenius endomorphism.

We first deal with the cases where G is of type SL_{n}, where n = 3 or n ≥ 5. Let s be an involution with centralizer of type A_{n−3}A_{1}. Then s is a semisimple element as required. (Again, see Table 4.5.1 of [3].)

For the remaining classical groups, the symplectic and orthogonal groups and SL_{2} and SL_{4}, first assume that p > 3. Let H be the centralizer in G^{∗} of the group of diagram automorphisms of the Dynkin-diagram of G^{∗}. Then H is again a reductive group of positive rank defined over 𝔽_{p}. Let A be an F_{0}-stable A_{1}-type subgroup of H. Then the fixed group A^{F_{0}} (which is isomorphic to SL_{2}(p) or PGL_{2}(p)) contains a semisimple element s of order 3. Since elements of order 3 in these groups are real, s is a real element of A^{F_{0}}, hence of H^{F_{0}}, and hence of Ĝ^{∗} = (G^{∗})^{F}. Since G is simple of symplectic or orthogonal type or SL_{2} or SL_{4}, we see that 2 is the only bad prime for G, and |Z(G)| is of 2-power order. Thus s has connected centralizer in G^{∗} by Corollary 2.25 of [6]. Moreover, by construction, the class of s is invariant under the full automorphism group of G^{∗}, and thus the corresponding semisimple character χ_{s} is as required.

This leaves the case p = 3. If G is not of type A_{1}, then the centralizer of the group of diagram automorphisms of G^{∗} is of type B_{m} or C_{m} or G_{2}, where the latter possibility occurs when G is of type D_{4}. The centralizer therefore contains a subgroup of type B_{2} or of type G_{2} in all cases. (Note that we already excluded SL_{3}.) Now SO_{5}(3) contains a real semisimple element of order 5 and G_{2}(3) contains a real semisimple element of order 7 in their derived subgroups. (This can be checked in the Atlas [1].) In either case, this defines a semisimple conjugacy class [s] of real elements in G^{∗}. As above, 2 is the only bad prime and is the only prime dividing |Z(G)|, and so the centralizer of s in G^{∗} is connected. This yields the desired semisimple character χ_{s}.

We are left with SL_{2} over fields of characteristic 3, and so we have Ĝ = SL_{2}(3^{a}) for some integer a > 1. Suppose that a is not a power of 3 and write a = 3^{f}b, where b > 1 is coprime to 3. Then the centralizer in Ĝ^{∗} = P GL_{2}(3^{a}) of a field automorphism of maximal 3-power order is P GL_{2}(3^{b}). Take any odd-order semisimple element s ∈ P GL_{2}(3^{b})′ = L_{2}(3^{b}). (Such an element exists since not both 3^{b} − 1 and 3^{b} + 1 can be powers of 2 because b > 1). Since semisimple elements of P GL_{2}(3^{b}) are always real, we have a real semisimple character χ_{s} of Ĝ/Z(Ĝ) of 3′-degree, and this character is invariant under a Sylow 3-subgroup of Aut(G).

REFERENCES

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7. Y. Wang and Z. Chen, Solubility of finite groups admitting a coprime order operator group, Boll. Un. Mat. Ital. A, 7 (1993), 325–331.