Corresponding author: Dr. M. Hertweck, Universität Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart Fax: (49 711)685-5304 E-mail: hertweck@mathematik.uni-stuttgart.de

COLEMAN AUTOMORPHISMS OF FINITE GROUPS

MARTIN HERTWECK AND WOLFGANG KIMMERLE

Abstract. An automorphism σ of a finite group G whose restriction to any Sylow subgroup equals the restriction of some inner automorphism of G shall be called Coleman automorphism, named for D. B. Coleman, who’s important observation from [2] especially shows that such automorphisms occur naturally in the study of the normalizer N of G in the units U of the integral group ℤG. Let Out_{Col}(G) be the image of these automorphisms in Out(G). We prove that Out_{Col}(G) is always an abelian group (based on previous work of E. C. Dade, who showed that Out_{Col}(G) is always nilpotent).

We prove that if no composition factor of G has order p (a fixed prime), then Out_{Col}(G) is a p′-group. If O_{p}(G) = 1, it suffices to assume that no chief factor of G has order p.

If G is solvable and no chief factor of G/O_{2}(G) has order 2, then N = GZ, where Z is the center of U. This improves an earlier result of S. Jackowski and Z. Marciniak.

1. Introduction

Let G be a finite group. In this paper we study (under certain conditions on the composition factors of G) automorphisms of G whose restriction to any Sylow subgroup equals the restriction of some inner automorphism of G. These automorphisms form a characteristic subgroup Aut_{Col}(G) of Aut(G), and we shall call them Coleman automorphisms, in honour of Donald B. Coleman, who essentially observed the following [2].

Coleman Lemma. Let P be a p-subgroup of G. Let R be a commutative ring with pR ≠ R. Then N_{U(RG)}(P) = N_{G}(P) · C_{U(RG)}(P).

(In this form, it appears first in [28, Proposition 1.14]; see also [20, Theorem 2.6].) It means that a unit u ∈ U(RG) of the group ring RG normalizing P acts by conjugation on P like a group element g ∈ G. (We like to point out that the proof exploits the fact that G acts on the support of u. Thus the Coleman Lemma applies to infinite groups as well, in the obvious way.)

This observation has far reaching consequences for automorphisms of integral group rings (see [25, 31]). We do not go into details, but explain how Coleman automorphisms arise in this context.

Date: May 23, 2000.

2000 Mathematics Subject Classification. Primary 20E36, 16U70; Secondary 20C10.

The research was partially supported by the Deutsche Forschungsgemeinschaft.

Recall that RG is an integral group ring if R is a G-adapted ring, that is, an integral domain of characteristic 0 in which no prime divisor of |G| is invertible. For a commutative ring R, let Aut_{R}(G) be the group of automorphisms of G which induce an inner automorphism of RG. Note that Aut_{R}(G) ≤ Aut_{ℚ}(G) = Aut_{c}(G), the latter being the group of class-preserving automorphisms (cf. [20, Proposition 2.5]). Hence we shall assume from now on that R is G-adapted, aiming for a nice blending of group theory and ring theory.

Then Coleman’s result shows that Aut_{R}(G) ≤ Aut_{Col}(G). Therefore Coleman automorphisms occur naturally in the study of the normalizer N_{U(RG)}(G) of G in the units U(RG) of RG, a subject of research initiated by the questions [20, 3.7] and [32, Problem 43]. To say what it is all about, let Out_{R}(G) = Aut_{R}(G)/Inn(G) (changing ‘A’ to ‘O’ will always have this fixed meaning). Then until recently, no groups G were known with Out_{R}(G) ≠ 1, or even Out_{ℤ}(G) ≠ 1 (see [15, 26]). On the other hand, it has turned out that groups with Out_{R}(G) ≠ 1 may give rise to counter-examples to the “isomorphism problem for integral group rings” (see [15, 23]). We shall say that a group G has the normalizer property (NP) if Out_{ℤ}(G) = 1, that is, if all units in ℤG normalizing G are ‘obvious’ ones:

N_{U(ℤG)}(G) = G · Z(U(ℤG)). (NP)

This may as well be understood as the normalizer problem, i.e. the problem to ‘determine’ in some sense the groups G with (NP).

To be honest, we do not know of any class of groups with (NP) for which this property cannot be verified entirely group-theoretical. Since a basic result of J. Krempa [20, Theorem 3.2] tells us that Out_{ℤ}(G) is an elementary abelian 2-group, one just has to show that Out_{Col}(G) ∩ Out_{c}(G) is a 2′-group. In this way, S. Jackowski and Z. Marciniak [20, Theorem 3.6] proved that (NP) holds for a finite group with normal Sylow 2-subgroup. It turns out that this is a special case of Corollary 3 below on Coleman automorphisms: If G has a normal Sylow p-subgroup, then Out_{Col}(G) is p′-group.

Besides this result, the impact of the structure of composition factors of G, and their position in a normal series of G on the groups Out_{Col}(G) and Out_{ℤ}(G) is poorly understood. The aim of this paper is to show that positive results for an arbitrary finite group are possible if we impose restrictions on the dimension of the abelian composition factors (see [21] for a first attempt in this direction).

With respect to non-abelian composition factors, the following hold. Let G be a simple group. Then W. Feit and G. M. Seitz [10, Theorem C] proved that Out_{c}(G) = 1. It follows that (NP) holds for simple groups. We show that also Out_{Col}(G) = 1. More generally, we show that for some prime p, the p-central automorphisms of G are inner automorphisms (Theorem 14). (A p-central automorphism of a finite group G is an automorphism which centralizes a Sylow p-subgroup of G.) All these results depend on the classification of the finite simple groups.

The result of Jackowski and Marciniak is a result for solvable groups, by the Odd Order Theorem. Let p be a fixed prime. We prove that if G is p-constrained and no chief factor of G/O_{p}(G) is of order p, then Out_{Col}(G) is a p′-group (Corollary 20). In particular, (NP) holds for G if G is solvable and G/O_{2}(G) has no chief factor of order 2, which generalizes the result of Jackowski and Marciniak.

We also prove that if the center of the generalized Fitting subgroup F^{∗}(G) is a p′-group, and if G/F^{∗}(G) has no chief factor of order p, then Out_{Col}(G) is a p′-group (Theorem 21).

In [24, p. 176], M. Mazur conjectured, with frankly very little supporting evidence, that Out_{ℤ}(G) = 1 for finite groups G with abelian Sylow 2-subgroups. It is true if in addition G is metabelian. Again, this is the consequence of a grouptheoretical fact, and the before mentioned result of Krempa. Namely, it has been proved in [14, 16] that if G is a metabelian group with abelian Sylow p-subgroups, then Out_{c}(G) is a p′-group. If G has abelian Sylow p-subgroups, and G/F^{∗}(G) has no chief factor of order p, then Out_{Col}(G) is a p′-group (Corollary 22, which follows from Theorem 21).

Further, we remark that there are groups G whose Sylow subgroups are all abelian, and Out_{Col}(G) ∩ Out_{c}(G) ≠ 1 [16, Example 1]. So if Mazur’s conjecture would be true, some not yet discovered ring-theoretical aspects should play a role. Finally, we would like to mention that there are finite metabelian groups G with Out_{ℤ}(G) ≠ 1 [15].

In Section 3 we prove that Out_{Col}(G) is always abelian. At this place, we explain some of the background.

Let K be a collection of subgroups of G. The automorphisms σ of G with the property that for any K ∈ K, there is τ ∈ Aut(G) centralizing K in the common sense such that σ and τ differ by an inner automorphism form a group, and their image in Out(G) is suggestively denoted by C_{Out(G)}(K).

We remark that if K is the family Syl_{∗}(G) of all Sylow subgroups of G, then C_{Out(G)}(K) is the group Out_{Col}(G) in which we are interested in. If K is the family Cyc(G) of all cyclic subgroups of G, then C_{Out(G)}(K) is the group Out_{c}(G). Let Nil(G) be the family of all nilpotent subgroups, and Ab(G) the family of all abelian subgroups.

In a series of papers [4–9], E. C. Dade investigated the subgroups C_{Out(G)}(K), with K being Nil(G), Ab(G) or Syl_{∗}(G). His motivation came from Note B in Burnside’s book [1], which states that Out_{c}(G) is always abelian. However, this is wrong; first counterexamples were given by C. H. Sah [27] (who also showed, using Schreier’s hypothesis, that Out_{c}(G) is always solvable). The problem remained to find a ‘good substitute’. Among other things, Dade proved that C_{Out(G)}(Syl_{∗}(G)) is nilpotent, even abelian if G is solvable. He also gave an example [5, 2.3] showing that C_{Out(G)}(Nil(G)) can be any finite abelian group. Finally, he remarked that it is “reasonable to suppose that C_{Out(G)}(Syl_{∗}(G)) is always abelian” [9, Corollary, p. 57].

Using the fact that Out_{Col}(G) is nilpotent, and results on p-central automorphisms by G. Glauberman (p = 2) and F. Gross (p > 2), we are able to prove that Out_{Col}(G) is always abelian.

Our notation is mostly standard. For group elements x, y we set x^{y} = y^{−1}xy and [x, y] = x^{−1}x^{y}. By conj(y) we denote any homomorphism of the form x ↦ x^{y}.

2. Preliminary Results

Throughout this section, G denotes a finite group. We list some useful results on class-preserving and Coleman automorphisms of G.

Proposition 1. The prime divisors of |Aut_{c}(G)| and |Aut_{Col}(G)| lie in π(G), the set of prime divisors of |G|.

Proof. It is known that prime divisors of the order of Aut_{c}(G) lie in π(G) (see [18, I.4 Aufgabe 12]). Let σ ∈ Aut_{Col}(G), and assume that σ has order r, with (r, |G|) = 1; we have to show that σ = id. Let p ∈ π(G). Since the action of σ on G is coprime, there is a Sylow p-subgroup P of G which is fixed by σ. By assumption, there is x ∈ G with σ|_{P} = conj(x)|_{P}. So σ induces an automorphism of P whose order divides r and the order of x. It follows that σ|_{P} = id|_{P}. As p ∈ π(G) was chosen arbitrary, it follows that σ = id.

Lemma 2. Let α ∈ Aut(G) be of p-power order, for some prime p. Assume that there is N ⊴ G such that Nα = N, and that α induces the identity on G/N. Let U be a subgroup of G. Then, if there is h ∈ G such that gα = g^{h} for all g ∈ U, there is n ∈ N with gα = g^{nh_{p}} for all g ∈ U, where h_{p} denotes the p-part of h.

Proof. The proof consists of a straightforward calculation. Let q be a power of p such that α^{q} = id, and h^{q} is the p′-part of h. Then for all g ∈ U,

g = gα^{q} = g^{k} with k = h(hα)(hα^{2}) … (hα^{q−1}).

Since α induces the identity on G/N, there is n ∈ N with k = h^{q}n^{−1}. It follows that gα = g^{h^{q}h_{p}} = g^{(h^{q}n^{−1})(nh_{p})} = g^{k(nh_{p})} = g^{nh_{p}} for all g ∈ U.

As an application, we have the following corollary (extending some results from [22, section 2] considerably).

Corollary 3. Let N ⊴ G and let p be a prime which does not divide the order of G/N. Then the following hold.

(i) If σ ∈ Aut(G) is a class-preserving or a Coleman automorphism of G of ppower order, then σ induces a class-preserving or a Coleman automorphism of N, respectively;

(ii) If Out_{c}(N) or Out_{Col}(N) is a p′-group, then so is Out_{c}(G) or Out_{Col}(G).

Proof. (i) By Proposition 1, σ induces the identity on G/N, and the statement follows immediately from Lemma 2.

(ii) Assume that Out_{c}(N) or Out_{Col}(N) is a p′-group, and let σ be a classpreserving or a Coleman automorphism of G, of p-power order. We have to show that σ ∈ Inn(G). By (i), and the assumption, σ induces an inner automorphism conj(x) of N. Replacing σ by a suitable power of σ conj(x^{−1}) (see Remark 5 below), we may assume that σ|_{N} = id|_{N}. By Proposition 1, σ induces the identity on G/N. It follows that σ induces the identity on G/M, where M = O_{p}(Z(N)), so σ corresponds to an element of H^{1}(G/M, M). But σ fixes a Sylow p-subgroup (which is contained in N) element-wise, and it follows that σ is an inner automorphism (cf. [18, I 16.18]).

The last argument in the proof, which will be used few times more, has been used already by F. Gross [13, Proposition 8.1] to prove the following proposition. Recall that G is p-constrained if CḠ(O_{p}(Ḡ)) ≤ O_{p}(Ḡ) for Ḡ = G/O_{p′}(G).

Proposition 4. Let G be a p-constrained group with O_{p′}(G) = 1, for some prime p. Then p-central automorphisms of G are inner automorphisms, given by conjugation with elements from Z(O_{p}(G)).

Finally, the following remark will prove to be very useful, and will be applied throughout the paper.

Remark 5. If σ ∈ Aut(G) is of p-power order, for some prime p, and if there are U ≤ G and x ∈ G with σ|_{U} = conj(x)|_{U}, then there is γ ∈ Inn(G) such that σγ|_{U} = id, and the order of σγ is still a power of p (taking for σγ a suitable power of σ conj(x^{−1})). In proofs, this fact will be used several times without any further comment, just indicated by a phrase like “we modify σ (by an inner automorphism)”.

3. Out_{Col}(G) is Abelian

Let G be a finite group. In this section, we prove that Out_{Col}(G) is an abelian group. We begin with a basic observation.

Lemma 6. Let σ ∈ Aut(G) and M ⊴ G with Mσ = M, and suppose that for some Sylow subgroup Q of M, there is h ∈ G such that σ|_{Q} = conj(h)|_{Q}. Then σ stabilizes N = MC_{G}(Q) ⊴ G, and σ induces on Ḡ = G/N the inner automorphism conj(h̄).

Proof. By the Frattini argument, N = MC_{G}(Q) is a normal subgroup of G, and by Sylow’s theorem, there is m ∈ M with Q^{hm} = Q. Let c ∈ C_{G}(Q). Then M(cσ) ∋ (cσ)^{m} ∈ C_{G}((Qσ)^{m}) = C_{G}(Q), showing that σ stabilizes N. Take any g ∈ G and x ∈ Q. By Sylow’s theorem, there is n ∈ M with Q^{gn} = Q, and it follows that x^{gnh} = (x^{gn})σ = x^{h(gσ)(nσ)}, so h(gσ)(nσ)h^{−1}n^{−1}g^{−1} ∈ C_{G}(Q) and h(gσ)h^{−1}g^{−1} ∈ N. That is, gσ ∈ Ng^{h} for all g ∈ G, and the lemma is proved.

Lemma 7. Let P be a Sylow p-subgroup of G, fixed by some σ_{1}, σ_{2} ∈ Aut(G). Assume that σ_{1} is of p-power order, and that σ_{2} fixes P element-wise. Let M ⊴ G with Mσ_{i} = M, and suppose that for some Sylow subgroup Q of M, there are h_{i} ∈ G such that σ_{i}|_{Q} = conj(h_{i})|_{Q}. Then [h_{1}, h_{2}] ∈ MC_{G}(Q).

Proof. By Lemma 6, N = MC_{G}(Q) ⊴ G is fixed by the σ_{i}, and σ_{i} induces the inner automorphism σ̄_{i} = conj(h̄_{i}) on Ḡ = G/N. Since σ̄_{1} is of p-power order, there is a pelement x ∈ G such that σ̄_{i} = conj(x̄). It follows that x̄ lies in the Sylow p-subgroup P̄ = P N/N of Ḡ, and that h̄_{1}^{−1}x̄ ∈ Z(Ḡ). Hence [h̄_{1}, h̄_{2}] = [x̄, h̄_{2}] = x̄^{−1}(x̄σ̄_{2}) = 1, which proves the lemma.

Corollary 8. Let P be a Sylow p-subgroup of G, fixed by some σ_{1}, σ_{2} ∈ Aut_{Col}(G). Assume that σ_{1} is of p-power order, and that σ_{2} fixes P element-wise. Let κ = [σ_{1}, σ_{2}]. Then κ|_{M} ∈ Aut_{Col}(M) for any M ⊴ G. Moreover, L = M{g ∈ G : conj(g)|_{M} ∈ Aut_{Col}(M)} is a normal subgroup of G, and κ induces the identity on G/L.

Proof. For any prime q, let S_{q} be a Sylow q-subgroup of M. By assumption, there are h_{i} ∈ G such that σ_{i}|_{S_{q}} = conj(h_{i})|_{S_{q}}. It follows from Lemma 7 that [h_{1}, h_{2}] ∈ N_{q} = MC_{G}(S_{q}). By Sylow’s theorem, σ_{i} restricted to any Sylow qsubgroup of M is given by conjugation with some element of the coset Mh_{i}. Hence κ|_{S_{q}} = conj(n) for some n ∈ M[h_{1}, h_{2}] ⊆ N_{q}. This shows that κ|_{M} ∈ Aut_{Col}(M). Moreover, κ induces the identity on G/N_{q} by Lemma 6. Since ∩_{q}N_{q} = L, it follows that σ induces the identity on G/L.

Corollary 9. Let σ_{1}, σ_{2} ∈ Aut_{Col}(G), and let M ⊴ G. Then there is γ ∈ Inn(G) such that [σ_{1}, σ_{2}]γ|_{M} ∈ Aut_{Col}(M).

Proof. Elementary commutator calculus shows that we may assume that the σ_{i} satisfy the conditions of Corollary 8, from which then the statement follows.

We make a short digression to solvable groups, in giving an alternative proof of the following result of Dade [4, Proposition 2.2]. The proof has been designed so that it can be taken over for a proof of the general result. (which, however, uses deep results of G. Glauberman and F. Gross on p-central automorphisms).

Corollary 10. Let G be a finite solvable group. Then Out_{Col}(G) is abelian.

Proof. Let A = Aut_{Col}(G), and put Ā = A/Inn(G). We first show that Ā is nilpotent. Let F = F(G) be the Fitting subgroup of G. Since F is nilpotent, Out_{Col}(F) = 1, and it follows from Corollary 9 that the commutator subgroup A′ is contained in Inn(G)B, where B = ⟨β ∈ Aut(G) : β|_{F} = id|_{F} ⟩. Let α ∈ A and β ∈ B; we show that [α, β] ∈ Inn(G). Clearly, it suffices to assume that β is of p-power order, for some prime p. Since C_{G}(F) ≤ F [18, III 4.2], it follows that β induces the identity on G/Z, where Z = O_{p}(Z(F)). Also, we may assume that α fixes some Sylow p-subgroup P of G element-wise. But then [α, β] fixes P elementwise, and induces the identity on G/Z, so [α, β] ∈ Inn(G) by [18, I 16.18]. Thus we have shown that Ā is nilpotent.

In order to prove that Ā is abelian, it therefore suffices to show that κ = [σ_{1}, σ_{2}] ∈ Inn(G), where σ_{1}, σ_{2} ∈ Aut_{Col}(G) are of p-power order. Modifying the σ_{i} by inner automorphisms, we may assume that they fix some Sylow p-subgroup P of G element-wise. Let N = O_{p′}(G). By Proposition 4, κ induces the identity on G/N. Since κ|_{N} ∈ Aut_{Col}(N) by Corollary 8, it follows from Proposition 1 that κ|_{N} is of p′-order. Hence κ is of p′-order. But κ̄ ∈ Ā is of p-power order since Ā is nilpotent. Hence κ ∈ Inn(G), which finishes the proof.

Let G be any finite group with O_{p′}(G) = 1 for some prime p, and let C be the group of automorphisms that fix every element of a (fixed) Sylow p-subgroup of G. If p = 2, G. Glauberman has proved that C has an abelian Sylow 2-subgroup and a normal 2-complement [11, Theorem 1]. Using the classification of the finite simple groups, F. Gross has proved that if p > 2, then C is even the direct product of a p′-group and an abelian p-group [13, Theorem A]. Finally, Dade has proved that Out_{Col}(G) is nilpotent [9, Corollary]. Using these results, we can prove the main theorem of this section.

Theorem 11. The normal subgroup Out_{Col}(G) of Out(G) is abelian, for every finite group G.

Proof. Fix some prime p. Since Out_{Col}(G) is nilpotent [9, Corollary], it suffices to show that κ = [σ_{1}, σ_{2}] ∈ Inn(G), where σ_{1}, σ_{2} ∈ Aut_{Col}(G) are of p-power order. Modifying the σ_{i} by inner automorphisms, we may assume that the σ_{i} fix some Sylow p-subgroup P of G element-wise. Let N = O_{p′}(G). Then κ induces on G/N an automorphism of p′-order. This follows from [11, Theorem 1] for p = 2, and from [13, Theorem A] for p > 2. Since κ|_{N} ∈ Aut_{Col}(N) by Corollary 8, it follows from Proposition 1 that κ|_{N} is of p′-order. Hence κ is of p′-order. On the other hand, by [9, Corollary], the image of σ_{i} in Out(G) lies in the Sylow p-subgroup of Out_{Col}(G), so the image of κ in Out(G) is of p-power order. It follows that κ ∈ Inn(G), and the theorem is proved.

4. Quasinilpotent Groups

In this section, we show that Out_{Col}(G) = 1 for a quasinilpotent group G. For convenience of the reader, we recall that a group G is quasinilpotent if G is a central product of its Fitting subgroup F(G) and its layer E = E(G), which is generated by the components, i.e. the subnormal quasisimple subgroups of G. The quotient E/Z(E) is a direct product of non-abelian simple groups. The generalized Fitting subgroup F^{∗}(G) = F(G)E(G) of a group G is quasinilpotent, and each quasinilpotent group coincides with its generalized Fitting subgroup. Note that C_{G}(F^{∗}(G)) ≤ F^{∗}(G). As a general reference, we give [19, X §13].

The following proposition is straightforward.

Proposition 12. Let A_{n} be the alternating group on n letters, for some n ≥ 5. Then there is a prime p dividing |A_{n}| such that p-central automorphisms of G are inner automorphisms.

Proof. There are non-inner 2-central automorphisms of A_{n} if and only if n ≡ 2, 3 mod 4 (this has been noticed already in [13, p. 202]). Hence we may assume that there is an odd prime p dividing n. Assume that conjugation by some permutation c ∈ S_{n} induces a p-central automorphism of A_{n}. Then we may assume that c centralizes the elements g_{i} = ((ip + 1)(ip + 2) ⋯ (i + 1)p), 0 ≤ i ≤ m = (n − p)/p, which lie in a Sylow p-subgroup P of A_{n}, and it follows that c ∈ C_{S_{n}}(P) ≤ P ≤ A_{n}. It remains to consider the case n = 6 and p = 3, but any exceptional automorphism of A_{6} exchanges the two conjugacy classes of elements of order 3.

With respect to simple groups of Lie type, we prove the following theorem.

Theorem 13. Let G be a finite group of Lie type, of characteristic p. Then pcentral automorphisms of G are inner automorphisms.

Proof. Let σ be p-central automorphism of G. By [29, Lemma 3], σ fixes each irreducible complex character belonging to the principal p-block of G. J. E. Humphreys [17] has shown that except the Steinberg character, each irreducible complex character belongs to the principal p-block. It follows that σ ∈ Aut_{c}(G), and therefore σ ∈ Inn(G) by [10, Theorem C].

We remark that a p-central automorphism σ of G even induces an inner automorphism of the principal p-block (a proof can be extracted from [29]). In particular, σ acts trivially on the cohomology ring H^{∗}(G, ℤ_{p}), a result which follows of course also from more elementary considerations.

Using the classification of the finite simple groups, we now can prove the following theorem.

Theorem 14. For any finite simple group G, there is a prime p dividing |G| such that p-central automorphisms of G are inner automorphisms.

Proof. According to the classification, and the above, only the sporadic groups remain to be inspected. Let K be one of the 12 sporadic groups with Out(K) = C_{2}. Direct inspection using the Atlas [3], or GAP [30], shows that the center of a Sylow 2-subgroup of Aut(K) is of order 2, so 2-central automorphisms of K are inner automorphisms.

Remark 15. 1. If G is a non-abelian simple group and p an odd prime, then p-central automorphisms of G of p-power order are inner automorphisms [13, Theorem A].

2. The theorem implies that for each non-abelian simple group G, there is a prime p such that for G ≤ H ≤ Aut(G), p-central automorphisms of H are inner automorphisms.

3. For G and H as above, one might ask whether Out_{c}(H) = 1. (This is true if H is simple by [10, Theorem C], and therefore also if Out(G) is cyclic.)

Corollary 16. Let G be a quasinilpotent group. Then Out_{Col}(G) = 1.

Proof. Let σ ∈ Aut_{Col}(G); we have to show that σ ∈ Inn(G). Let E = E(G) be the layer of G. We may assume that σ induces the identity on G/Z(E), by Theorem 14, and additionally that σ induces the identity on F(G). But this already implies that σ is the identity (since E is perfect).

Theorem 14 has the following corollary. Recall that a local subgroup of a finite group is a p-subgroup or the normalizer of a non-trivial p-subgroup.

Corollary 17. Let σ be an automorphism of a finite group G whose restriction to any local subgroup equals the restriction of some inner automorphism of G. Then σ ∈ Inn(G).

Proof. If G has a non-trivial solvable normal subgroup, then there is nothing to prove. Let M be a minimal normal subgroup of G. Then M = S_{1} × … × S_{t} with (isomorphic) simple groups S_{i}. By Theorem 14, there is a prime p ∈ π(M) such that p-central automorphisms of S_{i} are inner automorphisms. Let P be a Sylow psubgroup of M, and modify σ by an inner automorphism such that σ fixes N_{G}(P) element-wise. By the Frattini argument, G = MN_{G}(P), so σ induces on G/M the identity. By choice of p, we may modify σ again by an inner automorphism such that σ induces the identity on both G/M and M. It follows that σ is the identity.

5. When no Chief Factor of G/F^{∗}(G) is C_{p}

Let G be a finite group. In this section, Coleman automorphisms of G of p-power order (p a fixed prime) are studied under the assumption that no chief factor of G/F^{∗}(G) is isomorphic to C_{p}. This serves to sort out p-constrained groups (see Corollary 20 below) and is motivated by the following simple observation.

Lemma 18. Let σ ∈ Aut(G) be of p-power order. Let N_{1} and N_{2} be normal subgroups of G, fixed by σ, such that σ induces inner automorphisms of the G/N_{i}’s. If Ḡ = G/N_{1}N_{2} has no chief factor of order p, then σ induces an inner automorphism of G/N_{1} ∩ N_{2}.

Proof. By assumption, there are p-elements x_{i} ∈ G such that g^{−x_{i}}(gσ) ∈ N_{i} for all g ∈ G. Then z = x̄_{1}x̄_{2}^{−1} is a central element in Ḡ. It follows that z is a p-element, so z = 1 by assumption, and σ induces an inner automorphism of G/N_{1} ∩ N_{2}.

Lemma 6 allows us to prove the following positive results.

Lemma 19. Assume that no chief factor of G/F^{∗}(G) is isomorphic to C_{p}, and let σ ∈ Aut(G) be of p-power order. If σ induces the identity on G/N for some N ⊴ G with Nσ = N, and Q is a Sylow subgroup of N with σ|_{Q} = conj(x)|_{Q} for some x ∈ G, then there is g ∈ O_{p}(G)N with σ|_{Q} = conj(g)|_{Q}.

Proof. By Lemma 2, x̄ has p-power order in Ḡ = G/NC_{G}(Q). It follows from Lemma 6 that x̄ ∈ Z(Ḡ), and x ∈ F^{∗}(G)NC_{G}(Q) by the assumption on the chief factors. Hence x ∈ O_{p}(G)NC_{G}(Q).

Corollary 20. Let G be a p-constrained group, and assume that no chief factor of G/F^{∗}(G) is isomorphic to C_{p}. Then Out_{Col}(G) is a p′-group.

Proof. Let σ ∈ Aut_{Col}(G) be of p-power order; we have to show that σ ∈ Inn(G). Let N = O_{p′}(G). By Proposition 4, we may modify σ by an inner automorphism such that σ induces the identity on G/N. Then σ|_{N} ∈ Aut_{Col}(N) by Lemma 19. Hence σ|_{N}, and therefore σ too, is of p′-order, by Proposition 1. It follows that σ = id.

We do not know whether the assumption that G is p-constrained can be removed (see also the questions at the end of the section). We were able to prove the following theorem.

Theorem 21. Assume that no chief factor of G/F^{∗}(G) is isomorphic to C_{p}, and that Z(F^{∗}(G)) is a p′-group. Then Out_{Col}(G) is a p′-group.

Proof. Let σ ∈ Aut_{Col}(G) be of p-power order, and put F = F^{∗}(G). Note that if σ|_{F} = conj(g)|_{F} for some g ∈ G, then σ = conj(xg) for some x ∈ Z(F) since C_{G}(F) = Z(F) [19, X 13.12], which is a p′-group.

For any q ∈ π(F), let S_{q} be a Sylow q-subgroup of F and put N_{q} = F C_{G}(S_{q}). By Lemma 6, each N_{q} is a normal subgroup of G, and σ induces an inner automorphism of G/N_{q}. Since G/F has no chief factor isomorphic to C_{p}, it follows from Lemma 18 that σ induces an inner automorphism of G/N, where N = ∩_{q}N_{q}. Modifying σ by an inner automorphism, we may assume that σ induces the identity on G/N. Then σ|_{F} ∈ Aut_{Col}(F) by Lemma 19, and σ|_{F} ∈ Inn(F) by Corollary 16. By the introductory remark, this completes the proof.

Corollary 22. Assume that G has abelian Sylow p-subgroups, and that no chief factor of G/F^{∗}(G) is isomorphic to C_{p}. Then Out_{Col}(G) is a p′-group.

Proof. Let σ ∈ Aut_{Col}(G) be of p-power order; we have to show that σ ∈ Inn(G). By Theorem 21, we may assume that N = O_{p}(G) ≠ 1, and that σ induces the identity on G/N. Let P be a Sylow p-subgroup of G. Then P is fixed by σ, and there is a p-element x ∈ G with σ|_{P} = conj(x)|_{P}. It follows that x ∈ P, and, as P is abelian, that σ|_{P} = id|_{P}. Hence σ ∈ Inn(G).

We conclude this section with some questions which remain unanswered in this paper. Let G be a finite group, and p a fixed prime. The answer to these questions is yes if G is solvable, or, more generally, p-constrained (see Proposition 4 and Corollary 20).

Question. 1. Assume that no chief factor of G is of order p. Is it true that

Out_{Col}(G) is a p′-group?

2. Assume that O_{p′}(G) = 1 for some prime p. Is it true that Out_{Col}(G) = 1?

3. Assume that G has a unique minimal normal subgroup. Is it true that Out_{Col}(G) = 1?

We remark that F. Gross conjectured that if G is a finite group with O_{p′}(G) = 1 for some odd prime p, then any automorphism of G centralizing a Sylow p-subgroup of G and having order a power of p is an inner automorphism [13, p. 203]. Using the information available on Schur multipliers of simple groups, it should be possible to prove this conjecture.

If G has exactly one component L with M = O_{p}(Z(L)) ≠ 1, then the first question has a positive answer. Indeed, as M is not cyclic, it follows from the table given in [12] that M ≅ C_{2} × C_{2} or M ≅ C_{3} × C_{3}. Let K = C_{G}(M), a proper normal subgroup of G. If p = 2 and M ≅ C_{2} × C_{2}, then G/K has a chief factor of order 2 as GL(2, 2) ≅ S_{3}. If p = 3 and M ≅ C_{3} × C_{3}, then G/K is a 3′-group since GL(2, 3) is solvable with Sylow 3-subgroup of prime order (PSL(2, 3) ≅ A_{4}, see [18, II 6.14]). This forces G/K ≅ C_{2}. But then M contains a central subgroup of G of order 3, a contradiction.

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Mathematisches Institut B, Universität Stuttgart, Pfaffenwaldring 57, D-70550 Stuttgart, Germany

   E-mail address: hertweck@mathematik.uni-stuttgart.de
   E-mail address: kimmerle@mathematik.uni-stuttgart.de