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

CLASS-PRESERVING AUTOMORPHISMS OF FINITE GROUPS

MARTIN HERTWECK

Abstract. We construct a family F of Frobenius groups having abelian Sylow subgroups and non-inner, class-preserving automorphisms. We show that any A-group (that is, a finite solvable group with abelian Sylow subgroups) has a section belonging to F provided it has a non-inner, class-preserving automorphism. As a consequence, we obtain that for metabelian A-groups, or A-groups with elementary abelian Sylow subgroups, class-preserving automorphisms are necessarily inner automorphisms. The same is true for a finite group whose Sylow subgroups of odd order are all cyclic, and whose Sylow 2-subgroups are either cyclic, dihedral or generalized quaternion. Some applications are given.

1. Introduction

The influence of the existence of a group automorphism of a certain kind on the structure of a finite group is the subject of much research. In this paper, we survey class-preserving automorphisms of finite groups, that is, automorphisms that map each group element to some conjugate of it. For a finite group G, these automorphisms form a group, denoted by Aut_{c}(G). Clearly Aut_{c}(G) contains Inn(G), the group of inner automorphisms of G, and we put Out_{c}(G) = Aut_{c}(G)/Inn(G).

The natural question arises whether Out_{c}(G) may be non-trivial; it was posed in 1911 by W. Burnside in [3, Note B]. He also gave the answer in [4] when constructing a group G of order p^{6} with Out_{c}(G) ≅ C_{p}^{4}, where p is a prime with p ≡ ±3 mod 8. Since then, further examples have been produced, see for example [6,16,19].

Nowadays it is well-known that nilpotent groups are the most natural candidates for groups with non-inner, class-preserving automorphisms. One of the basic facts about class-preserving automorphisms is that primes dividing the order of Aut_{c}(G) also divide the order of G. It is therefore natural to look for a class of groups such that for members G with non-inner, class-preserving automorphisms of p-power order (for some prime p), and which are in a certain sense “minimal” with respect to this property, O_{p}(G) = 1 holds. Thus we are led to consider the class of Agroups, i.e. solvable groups whose Sylow subgroups are all abelian (cf. Lemma 3). Here, we obtain a quite satisfactory description. Theorem A, containing the main result of this paper, states that each A-group G with Out_{c}(G) ≠ 1, but Out_{c}(X) = 1 for proper subgroups or factor groups X of G, belongs to a certain class of Frobenius groups, which is described in Theorem A′. It follows that each A-group G with Out_{c}(G) ≠ 1 has a section belonging to this class. As further corollaries, we obtain that Out_{c}(G) = 1 if G is a metabelian A-group, or an A-group with elementary abelian Sylow subgroups.

Date: June 17, 1999.

1991 Mathematics Subject Classification. 20D45.

We like to mention that in connection with the isomorphism problem for integral group rings and with certain versions of the conjecture of Zassenhaus that group bases of ℤG are rationally conjugate, there are obstructions involving classpreserving automorphisms for the conjectures to be true (see [11]), and this is where the authors interest stems from. In Section 7.1 we briefly touch upon this subject. We establish a version of the conjecture of Zassenhaus for certain nilpotent by metabelian groups, and for Frobenius groups, the latter through the use of our Theorem B, which states that Out_{c}(G) = 1 for any finite group G whose Sylow subgroups of odd order are all cyclic, and whose Sylow 2-subgroups are either cyclic, dihedral or generalized quaternion. This extends results obtained by W. Kimmerle and K. Roggenkamp in [11].

The paper is organized as follows. In Section 2, the main theorems and some corollaries are given. Theorem A′, which describes a class of Frobenius groups G with Out_{c}(G) ≠ 1, is proved in Section 3, where also some concrete examples are given. Theorems A and B are proven in Sections 5 and 6, with the preparatory work done in section 4, where some properties of class-preserving automorphisms are collected, which are also of interest of their own. Section 7 is devoted to some applications. The obtained results are used to establish certain conjectures for integral group rings. Finally, using the examples given in Theorem A′, we show how to construct A-groups G with non-inner, class-preserving automorphisms which agree on each Sylow subgroup with an inner automorphism (Example 1), thus answering a question of M. Mazur.

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. Statement of the Results

We shall construct Frobenius groups G = G(p^{a−b}, r, n), parameterized by certain tuples (p^{a−b}, r, n) (see notation below). These groups are A-groups which have noninner, class-preserving automorphisms. They are described in Theorem A′, while Theorem A asserts that they are in a certain sense the “minimal” ones among the A-groups with non-inner, class-preserving automorphisms:

Theorem A. Let G be an A-group. Assume that G possesses a class-preserving, non-inner automorphism, but that class-preserving automorphisms of proper subgroups or factor groups are inner automorphisms. Then G is a group as described in Theorem A′ below.

These groups are subgroups of affine semi-linear groups, so let us recall their definition.

Definition 1. Let F be the Galois field 𝔽_{r^{m}} of order r^{m}, for some prime r. We write F^{×} for its unit group, and V for F, if considered as an additive group only. Let G = Gal(F/𝔽_{r}) be the Galois group of F over 𝔽_{r}. The affine semi-linear group AΓ(F) is the semidirect product V ⋊ F^{×} ⋊ G, with operations defined by v^{c} = vc, v^{σ} = vσ and c^{σ} = cσ for all v ∈ V, c ∈ F^{×} and σ ∈ G. Alternatively, AΓ(F) can be viewed as a group of permutations x ↦ c(xσ) + v on the set V, where σ ∈ G, c ∈ F^{×} and v ∈ V. Then we have the following subgroups:

1. A(F) = V = {x ↦ x + v | v ∈ V }, consisting of translations;

2. Γ(F) = F^{×} ⋊ G = {x ↦ c(xσ) | σ ∈ G, c ∈ F^{×}}, the semi-linear group;

3. Γ_{0}(F) = F^{×} = {x ↦ cx | c ∈ F^{×}}, consisting of multiplications.

In order to state Theorem A′, we fix some notation:

• p and r are different primes, and n ∈ ℕ satisfies p^{2}| r^{n} − 1;

• F = 𝔽_{r^{np}}, and V = (F, +);

• ϕ is the Frobenius automorphism of F;

• ⟨ω⟩ is the Sylow p-subgroup of F^{×}, of order p^{a}, and we fix b ∈ ℕ with 1 ≤ b ≤ a − 2 (it turns out that a ≥ 3);

• ⟨η⟩ is the subgroup of F^{×} of order e = \frac{r^{np}−1}{r^{n}−1}, and ξ = η^{p}.

Theorem A′. Keeping the above notation, let G = G(p^{a−b}, r, n) = ⟨V, ξ, ϕ^{n} · ω^{p^{b}}⟩ as a subgroup of the affine semi-linear group AΓ(F). Then the following holds.

1. G is a Frobenius group with Frobenius kernel V. A complement C to V is given by C = ⟨ξ⟩ ⋊ ⟨ϕ^{n} · ω^{p^{b}}⟩, of order e · p^{a−b−1}. The Frobenius kernel V is a simple ⟨η⟩-module.

2. The group of class-preserving automorphisms of G modulo the inner ones has order p and is generated by the image of an automorphism σ, defined by vσ = vω^{p^{b}} and cσ = c for all v ∈ V and c ∈ C.

We obtain the following corollaries.

Corollary 1. Let G be an A-group with a class-preserving, non-inner automorphism. Then some section of G is isomorphic to a group described in Theorem A′.

Recall that a section of G is a factor group U/N with N ⊴ U ≤ G. We omit the obvious proof.

Corollary 2. Class-preserving automorphisms of an A-group with elementary abelian Sylow subgroups are inner automorphisms.

This follows from the last corollary since any group G(p^{a−b}, r, n) contains elements of order p^{2}.

Looking carefully at the first part of the proof of Theorem A, the next three corollaries are obtained. They have already been recorded in [7, Satz 1].

Corollary 3. Class-preserving automorphisms of a metabelian A-group are inner automorphisms.

More precisely, the proof shows that

Corollary 4. If G is a metabelian group with abelian Sylow p-subgroups, then class-preserving automorphisms of G of order a power of p are inner automorphisms.

According to [15, p. 176], this result has been obtained recently by Marciniak and Roggenkamp in the special case p = 2.

Corollary 5. Let G be a group which is an extension of an abelian q-group by an abelian q′-group, for some prime q. Then any automorphism of G which preserves the conjugacy classes of elements of prime power order is an inner automorphism.

We are also dealing with another class of groups, which comprises the class of Frobenius complements, and prove the following theorem.

Theorem B. Let G be a finite group whose Sylow subgroups of odd order are all cyclic, and whose Sylow 2-subgroups are either cyclic, dihedral or generalized quaternion. Then each class-preserving automorphism of G is an inner automorphism.

3. Proof of Theorem A′ and some Examples

Proof of Theorem A′. Let E = 𝔽_{r^{n}} ⊂ F = 𝔽_{r^{np}}; then [F : E] = p and Gal(F/E) = ⟨ϕ^{n}⟩. Since p^{2}| r^{n} − 1,

\frac{e}{p} = \frac{r^{np} − 1}{p(r^{n} − 1)} = 1 + \frac{r^{n} − 1}{p} ∑_{i=1}^{p−1} \frac{r^{ni} − 1}{r^{n} − 1} ∈ ℕ  (1)

and

gcd(\frac{e}{p}, p) = 1, gcd(\frac{e}{p}, r^{n} − 1) = 1.  (2)

(This also follows from [12, VI 6.2(ii)].) By definition of e,

[F^{×} : E^{×}] = e.  (3)

Recall that ⟨ω⟩ is the Sylow p-subgroup of F^{×}, of order p^{a}, that ξ is an element of order \frac{e}{p} in F^{×}, and that C = ⟨ξ, ϕ^{n} · ω^{p^{b}}⟩ is a complement to V in G. Clearly, ϕ^{n} · ω^{p^{b}} has order p^{a−b}, so C = ⟨ξ⟩ ⋊ ⟨ϕ^{n} · ω^{p^{b}}⟩ has order e · p^{a−b−1}. Since p^{2}| r^{n} − 1,

(1) and (3) imply that

a ≥ 3.  (4)

G is a Frobenius group, with Frobenius kernel V: Otherwise, there are 0 ≠ v ∈ V and ξ^{i} ∈ ⟨ξ⟩ such that in G, the element v is centralized by ξ^{i} · ϕ^{n} · ω^{p^{b}} ∈ C, i.e. v = (vξ^{i})ϕ^{n} · ω^{p^{b}} or

v^{−1}(vϕ^{n}) = (ξ^{−i}ϕ^{n})ω^{−p^{b}}.

Note that F^{×} → F^{×}, x ↦ x^{−1}(xϕ^{n}) is a group homomorphism with kernel E^{×}. By (3), the image has order e. By (2), the element η := ξω^{p^{a−1}} has order e, too, so

{x^{−1}(xϕ^{n}) | x ∈ F^{×}} = ⟨η⟩.  (5)

By (2) and (4), the p-part of v^{−1}(vϕ^{n}) has order p^{a−b}, which is ≥ p^{2} by (4), whereas the p-part of η has order p. We have reached a contradiction.

The automorphism σ of G is defined by vσ = vω^{p^{b}} and cσ = c for all v ∈ V and c ∈ C (since ω^{p} ∈ E, the fixed field of ϕ^{n}). We show that σ is a class-preserving automorphism: As G is a Frobenius group, [·, c] : V → V is a bijection for every c ∈ C, and it suffices to show that v is conjugate to vσ for every v ∈ V \{0}. Take any v ∈ V \ {0}. Note that η = ξ · (ω^{p^{b}} · ϕ^{n})^{p^{a−b−1}} ∈ C. By (5), there is λ ∈ ⟨η⟩ with (vσ)^{−1}(vσϕ^{n}) = λ. It follows that

v^{ω^{p^{b}}·ϕ^{n}·λ^{−1}} = vσ  (6)

(in G), as desired.

Assume that σ is an inner automorphism. Then there is c ∈ C with σ|_{V} = conj(c)|_{V}. It follows that [C, c] ≤ C_{C}(V) = 1 and c ∈ Z(C) = ⟨ω^{p^{b+1}}⟩, a contradiction. Hence σ is a non-inner automorphism. Clearly σ^{p} = conj(ω^{p^{b+1}}) ∈ Inn(G).

V is a simple ⟨η⟩-module: Let 1 ≠ M ≤ V be a minimal normal subgroup of V ⋊ ⟨η⟩. We may assume without loss of generality that 1_{F} ∈ M, and therefore have to show that L := 𝔽_{r}(η) = F.

Note that L contains the primitive p-th root of unity ω^{p^{a−1}}. If ω ∉ L, then it follows from [12, VI 6.2(ii)] that the index [F : L] is divisible by p, whence L ⊆ E as [F : E] = p. But this is impossible since ξ ∈ L, and ξ ∉ E by (2). Hence ω ∈ L.

Let {x_{1}, …, x_{s}} be a basis of V, viewed as a vector space over L, and put M_{i} = x_{i}L. By (5), and as ω ∈ L, each M_{i} is a normal subgroup of G, and V = M_{1}×…×M_{s}. Assume that s > 1. Take c ∈ C with (x_{1} ⋯ x_{s})σ = (x_{1} ⋯ x_{s})^{c}. Then x_{i}σ = x_{i}^{c} for all i, and as G is a Frobenius group, it easily follows that σ|_{V} = conj(c)|_{V}. But this is impossible as we have already seen above. Hence s = 1, i.e. F = L, and we are done.

Finally, assume that β is any class-preserving automorphism of G; we have to show that β ∈ ⟨σ, Inn(G)⟩. Since class-preserving automorphisms of the metacyclic Frobenius complement C are inner automorphisms, and all Frobenius complements in G are conjugate, we may assume that β fixes each element of C. By (6), we may modify β by a suitable power of σ such that vβ = v^{c} for some v ∈ V \ {0} and c ∈ ⟨η⟩ ≤ C. Since V is a simple ⟨η⟩-module, it follows that β|_{V} = conj(c)|_{V}, and this implies that β is an inner automorphism.

Let us look at some examples. F denotes the family of groups G(p^{a−b}, r, n) described in Theorem A′.

1. G(2^{2}, 5, 1) is the smallest group belonging to F, of order 300. A presentation is given by

G(2^{2}, 5, 1) = ⟨ v, w, a, b : v^{5} = w^{5} = [v, w] = 1, a^{3} = b^{4} = 1, a^{b} = a^{2},
v^{a} = v^{2}w, w^{a} = v^{3}w^{2}, v^{b} = v^{3}, w^{b} = w^{2} ⟩.

In other words, the Frobenius kernel V = ⟨v, w⟩ is isomorphic to 𝔽_{5} ⊕ 𝔽_{5}, and a acts on V via the matrix [ \begin{matrix}2 & 1 \\ 3 & 2 \end{matrix} ] ∈ GL(2, 5), and b via [ \begin{matrix}3 & 0 \\ 0 & 2 \end{matrix} ] ∈ GL(2, 5).

2. 19 groups of order ≤ 10^{6} belong to F, with p = 2 for all of them. They are listed in the following table:

\begin{matrix}p^{a−b}, r, n & order of G(p^{a−b}, r, n) \\ 2^{2}, 5, 1 & 2^{2} · 5^{2} · 3 = 300 \\ 2^{2}, 3, 2 & 2^{2} · 3^{4} · 5 = 1620 \\ 2^{3}, 3, 2 & 2^{3} · 3^{4} · 5 = 3240 \\ 2^{2}, 13, 1 & 2^{2} · 13^{2} · 7 = 4732 \\ 2^{2}, 17, 1 & 2^{2} · 17^{2} · 3^{2} = 10404 \\ 2^{3}, 17, 1 & 2^{3} · 17^{2} · 3^{2} = 20808 \\ 2^{2}, 5, 2 & 2^{2} · 5^{4} · 13 = 32500 \\ 2^{4}, 17, 1 & 2^{4} · 17^{2} · 3^{2} = 41616 \\ 2^{2}, 29, 1 & 2^{2} · 29^{2} · 3 · 5 = 50460 \\ 2^{3}, 5, 2 & 2^{3} · 5^{4} · 13 = 65000 \end{matrix} \begin{matrix}p^{a−b}, r, n & order of G(p^{a−b}, r, n) \\ 2^{2}, 37, 1 & 2^{2} · 37^{2} · 19 = 104044 \\ 2^{2}, 41, 1 & 2^{2} · 41^{2} · 3 · 7 = 141204 \\ 2^{2}, 7, 2 & 2^{2} · 7^{4} · 5^{2} = 240100 \\ 2^{3}, 41, 1 & 2^{3} · 41^{2} · 3 · 7 = 282408 \\ 2^{2}, 53, 1 & 2^{2} · 53^{2} · 3^{3} = 303372 \\ 2^{2}, 61, 1 & 2^{2} · 61^{2} · 31 = 461404 \\ 2^{3}, 7, 2 & 2^{3} · 7^{4} · 5^{2} = 480200 \\ 2^{2}, 73, 1 & 2^{2} · 73^{2} · 73 = 788692 \\ 2^{4}, 7, 2 & 2^{4} · 7^{4} · 5^{2} = 960400 \end{matrix}

On a large scale, the distribution of the group orders of members of F shows a considerable regularity. In the diagram on the right, f(n) is the number of groups in F of order ≤ n.

[figure: 10^{15}; 10^{14}; 1000; 5000; f(n); n]

3. G(3^{2}, 19, 1), of order 19^{3} · 127 · 3^{2} = 7839837, is the smallest group of odd order in F.

4. Among the groups of type G(5^{a−b}, r, n), the group G(5^{2}, 101, 1) is the smallest one, having order 101^{5} · 31 · 491 · 1381 · 5^{2}.

5. G(3^{2}, 2, 6), of order 2^{18} · 3^{2} · 1387 = 3272343552, is the smallest group in F with Frobenius kernel of even order.

4. Preparations for the Proof of Theorems A and B

In this section, we collect some basic properties of class-preserving automorphisms of a finite group G, most of them connected with the existence of a proper normal subgroup of G. The group of class-preserving automorphisms of G is denoted by Aut_{c}(G), and we put Out_{c}(G) = Aut_{c}(G)/Inn(G).

One of the basic facts is that primes dividing the order of Aut_{c}(G) also divide the order of G. (This “first theorem” on class-preserving automorphisms was given by W. Burnside in 1897 [2, §159 Theorem II], see also [9, I.4 Aufgabe 12].)

If σ ∈ Aut_{c}(G) and N ⊴ G, then σ fixes N, and induces a class-preserving automorphism of G/N. For the induced automorphism of N we shall write σ|_{N}, by slight abuse of notation.

We shall need the following lemma.

Lemma 1. Let p be a prime, and α ∈ Aut(G) of order a power of p. Assume further that there is N ⊴ G such that Nα = N, and 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.

The lemma may be used to improve results from [14, section 2]. For our purpose, the following corollary suffices.

Corollary 6. Let σ ∈ Aut(G) of order a power of a prime p. If N ⊴ G such that p does not divide the order of G/N, the following holds.

(i) If σ ∈ Aut_{c}(G), then σ|_{N} ∈ Aut_{c}(N); (ii) If Out_{c}(N) is a p′-group, then so is Out_{c}(G).

Proof. Let σ ∈ Aut_{c}(G). As σ and G/N have coprime orders, σ induces the identity on G/N, and (i) follows immediately from Lemma 1.

Assume that Out_{c}(N) is a p′-group, and let σ ∈ Aut_{c}(G) be of order a power of p. By (i) and the assumption, we may modify σ by an inner automorphism to get σ|_{N} = id. Again, 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) elementwise, and it follows that σ is an inner automorphism [9, I 16.21].

The following observation is trivial, but will nevertheless play an important rôle.

Remark 1. Let σ ∈ Aut(G) be a class-preserving, non-inner automorphism, of order a power of a prime p, and γ ∈ Inn(G). Then (γσ)^{r}, where r denotes the p′- part of the order of γσ, is a class-preserving, non-inner automorphism of p-power order, too (obvious since ⟨σ^{r}⟩ = ⟨σ⟩ and (γσ)^{r} and σ^{r} differ only by an inner automorphism).

This will be applied in two ways. Let σ be a non-inner (class-preserving) automorphism of G, of order a power of p. Assume that

(i) there are U ≤ G and β ∈ Inn(G) with σ|_{U} = β|_{U}, or that

(ii) there is N ⊴ G with Nσ = N such that σ induces an inner automorphism on G/N.

Then there is γ ∈ Inn(G) such that (i) σγ|_{U} = id or (ii) σγ induces the identity on G/N, and σγ is still a non-inner (class-preserving) automorphism of order a power of p. Sometimes we will use this fact without any further comment, just saying that we “modify σ by an inner automorphism”, when we intend to replace σ by σγ.

The following lemma provides a useful criteria for certain automorphisms to be class-preserving.

Lemma 2. Let G be a finite group. Assume that A is an abelian normal subgroup of G with complement H such that for all primes dividing the order of A, Sylow p-subgroups of G are abelian. Then for any ϕ ∈ Aut(G) with hϕ = h for all h ∈ H, the following two statements are equivalent.

(i) ϕ ∈ Aut_{c}(G); (ii) For all a ∈ A and h ∈ H satisfying [a, h] = 1 there exists x ∈ H such that

(ah)ϕ = (ah)^{x}.

Proof. By coprime action [9, III 13.4], there is for any g ∈ G a direct product decomposition (∗) A = C_{A}(g) × [A, g].

(i)⇒(ii). Let a ∈ A and h ∈ H with [a, h] = 1. By assumption, there exist b ∈ A, x ∈ H such that (aϕ) · h = (ah)ϕ = (ah)^{xb} = a^{x}[b, h^{−x}] · h^{x}. Hence

[h, x] = 1 and aϕ = a^{x}[b, h^{−x}].

It follows that aϕ ∈ C_{A}(h^{−1}) = C_{A}(h^{−x}). Clearly a^{x} ∈ C_{A}(h^{−x}) and [b, h^{−x}] ∈ [A, h^{−x}], so aϕ = a^{x} by (∗) (applied with g = h^{−x}). Consequently, (ah)ϕ = (ah)^{x}.

(ii)⇒(i). Let a ∈ A and h ∈ H. By (∗), we may write a = bc with b ∈ C_{A}(h^{−1}) and c ∈ [A, h^{−1}]. By assumption, (bh)ϕ = (bh)^{x} for some x ∈ H. Thus bϕ = b^{x} and h = hϕ = h^{x}. It follows that c^{−x} ∈ [A, h^{−1}]. Clearly cϕ ∈ [A, h^{−1}], so c^{−x}(cϕ) ∈ [A, h^{−1}]. As [·, h^{−1}] is a permutation on [A, h^{−1}] by (∗), there exists d ∈ [A, h^{−1}] such that c^{−x}(cϕ) = [d, h^{−1}], and it follows that (ah)^{xd} = (bch)^{xd} = b^{x}c^{x}[d, h^{−1}]h = (ah)ϕ.

Lemma 3. Let G be a finite group satisfying the following “minimal condition”: There is a class-preserving, non-inner automorphism σ of G of order a power of p, for some prime p, and class-preserving automorphisms of proper factor groups of G are inner automorphisms. Then the following hold.

(i) The Frattini subgroup Φ(G) is a p-group. (ii) If Sylow p-subgroups are abelian, then O_{p}(G) = 1. (iii) If O_{p}(G) = 1, and C_{G}(N) ≤ N for some 1 ≠ N ⊴ G, then σ|_{N} ≠ γ|_{N} for all

γ ∈ Inn(G). Note that we do not require that class-preserving automorphisms of proper subgroups of G are inner automorphisms.

Proof. The Frattini subgroup of a group is nilpotent [9, III 3.6]; hence if Φ(G) is not a p-group, then N = O_{p′}(Φ(G)) is a non-trivial normal subgroup of G contained in Φ(G), and σ and N have coprime orders. A theorem of P. Hall [9, III 3.18] asserts that σ does not induce the identity on G/N. However, this contradicts Remark 1 and the minimality of G, and we have proved (i).

Assume that Sylow p-subgroups are abelian, and that N = O_{p}(G) ≠ 1. Then we may assume that σ induces the identity on G/N. By coprime action, there is a Sylow p-subgroup P of G with P σ = P. Take any x ∈ P. By Lemma 1, xσ = x^{g} for some p-element g ∈ G. Then x^{g^{i}} = xσ^{i} ∈ P for all i ∈ ℕ as N ≤ Z(P), so ⟨x^{g^{i}}, g | i ∈ ℕ⟩ is a p-group. Hence xσ = x^{g} = x. Now the same 1-cohomology argument as in the proof of Corollary 6(ii) supplies the contradiction σ ∈ Inn(G), and (ii) is proved.

Assume that C_{G}(N) ≤ N for some 1 ≠ N ⊴ G, and that there is γ ∈ Inn(G) with σ|_{N} = γ|_{N} for some γ ∈ Inn(G). Then, modifying σ by an inner automorphism if necessary, we may assume that σ induces the identity on both N and G/N. It follows that σ induces the identity modulo O_{p}(Z(N)), so O_{p}(G) ≠ 1 and (iii) is proved.

Note that the Fitting subgroup F(G) of a solvable group G contains its own centralizer [9, III 4.2].

In the above situation, further information about O_{p′}(F(G)) is given by the following theorem (see [9, III 3.14, 4.4, 4.5]).

Theorem 1. Let N be a non-trivial nilpotent normal subgroup of a finite group G with N ∩ Φ(G) = 1. Then N is the direct product of minimal normal subgroups of G, and possesses a complement in G.

Proposition 1. Let σ ∈ Aut(G) be of order a power of a prime p, and assume that G is a split extension of an abelian normal p′-subgroup A. Then there exists a complement K to A in G with Kσ = K.

Proof. A theorem of Gaschütz [9, I 17.4] asserts that if A is an abelian normal subgroup of a group H, and if A ≤ G ≤ H with (|A|, [H : G]) = 1, then A has a complement in H if it has one in G. Apply this theorem with H = G⟨σ⟩ (semidirect product) to get a complement L to A in G⟨σ⟩. The number of conjugates of L in G⟨σ⟩ is not divisible by p as A is a p′-group. Hence we may assume that Lσ = L, and K = L ∩ G is the desired complement.

Finally, we shall need the following lemma to prove Theorem A. The author has learnt the proof from [1] (see also [13, 2.1, 2.2] and [9, II 3.11]). Since we didn’t find a suitable reference for all parts of it, we shall sketch the proof for the readers convenience.

Lemma 4. Let G be a finite group with a normal abelian self-centralizing subgroup A. Let k be a prime field of order r with (r, |G|) = 1 and M a faithful irreducible kG-module such that M_{A} is a homogenous kA-module. Then the corresponding semidirect product M ⋊G may be identified with a subgroup of an affine semi-linear group AΓ(F), with M = A(F), Γ_{0}(F) · G = Γ_{0}(F) · Gal(F/E) and A ≤ Γ_{0}(F). Here, F/E is an extension of fields of degree [G : A] such that F is obtained from k by adjoining a primitive |A|-th root of unity, and E is isomorphic to End_{kG}(M).

Proof. The irreducible module M belongs to a block B of the semisimple group algebra kG. Let F be the projection of kA onto B. By Wedderburn’s theorem, the finite commutative algebra F is the direct product of fields. Actually F is a field because M is a homogenous F -module, and as G acts faithfully on B, F is an extension of k by a primitive |A|-th root of unity. G acts by conjugation as a group of automorphisms G on F, say g ∈ G acts via σ_{g} ∈ Aut(F). If some g ∈ G acts trivial on F, then [g, A] ≤ C_{G}(M) = 1, so g ∈ A as A is self-centralizing. Hence G ≅ G/A. It follows that B is isomorphic to the crossed-product algebra ⊕_{σ∈G}F σ with trivial factor set (for the definition, see [5, 8.33]). The center E of

B is the fixed field F^{G}, and [F : E] = |G| by Artin’s theorem [12, VI 1.8], so dim_{E}(B) = |G|^{2}. On the other hand, E ≅ End_{kG}(M) and B is isomorphic to a full matrix ring over E. Hence dim_{E}(M) = |G| and dim_{F} (M) = 1. Fix an isomorphism θ : M → F of vector spaces with mθ = 1. Then it is easily checked that we may extend θ to an injective homomorphism M ⋊ G → AΓ(F) by letting gθ = σ_{g} · (mg)θ for all g ∈ G.

5. Proof of Theorem A

Let G be an A-group satisfying the hypothesis of Theorem A. Then G possesses a class-preserving, non-inner automorphism σ of order a power of p, for some prime p. When we talk about the minimality of G, we shall mean that classpreserving automorphisms of proper subgroups or factor groups of G are inner automorphisms.

(1) O_{p}(G) = 1.

Proof. Lemma 3(ii).

(2) The Fitting subgroup F(G) has a complement in G. Each normal subgroup A of G with A ≤ F(G) has a complement B in F(G) which is a normal subgroup of G.

Proof. Immediate from Lemma 3(i), (ii) and Theorem 1.

(3) G is not metabelian.

Proof. Assume that G is metabelian. Then F := F(G) and G/F are abelian groups, and σ induces the identity on G/F. Hence A := ⟨σ|_{F} , conj(g)|_{F} : g ∈ G⟩ is an abelian subgroup of Aut(F). By (2), F can be written as the direct product of minimal normal subgroups M_{i} of G, 1 ≤ i ≤ s. Choose m_{i} ∈ M_{i} \ {1} for each i, and let α ∈ C_{A}(m_{1} ⋯ m_{s}). As M_{i} is a simple module for the abelian group A, the factor group A/C_{A}(M_{i}) acts without fixed points on M_{i}, and it follows that

α ∈ ⋂_{i=1}^{s} C_{A}(m_{i}) = ⋂_{i=1}^{s} C_{A}(M_{i}) = 1.

Consequently, σ|_{F} = conj(x)|_{F} for any x ∈ G with (m_{1} ⋯ m_{s})σ = (m_{1} ⋯ m_{s})^{x}, and we may modify σ by an inner automorphism such that σ|_{F} = id holds. But F is a p′-group by (1), so we get the contradiction σ = id.

Up to now, we could have restricted investigations to the metabelian case, so Corollary 3 is proved (by induction on the group order).

(4) The commutator subgroup G′ is a Hall p′-subgroup of G.

Proof. By Corollary 6 and the minimalitiy of G, G/G′ is a p-group. It remains to prove that G′ ≤ N := O_{p′}(G). Let M be the normal subgroup of G above N with F(G/N) = M/N. Then, as the Fitting subgroup of a solvable group contains its own centralizer [9, III 4.2] and Sylow p-subgroups of G are abelian, M/N is isomorphic to a Sylow p-subgroup of G. Hence G/M is a p′-group. As G/G′ is a p-group, it follows that G = M and G′ ≤ N.

(5) Let K be a complement to F(G) in G. Then Z(K) = O_{p}(K).

Proof. Note that by (2), such a complement exists. By (4), K′ is a Hall p′-subgroup of K, so O_{p}(K) is a central subgroup of K. For any A-group, the intersection of its commutator subgroup and its center is trivial [9, VI 14.3]. Thus K′∩Z(K) = 1, completing the proof.

From now on, we assume that σ induces the identity on G/F(G). By Proposition 1, there is a complement K to F(G) in G with Kσ = K. Note that this implies that kσ = k for all k ∈ K, and that σ|_{F(G)} is a K-module homomorphism of F(G).

(6) There is no k ∈ K with σ|_{F(G)} = conj(k)|_{F(G)}.

Proof. This is shown by a direct calculation or application of Lemma 3(iii).

(7) F(G) is the unique minimal normal subgroup of G, henceforth denoted by M.

Proof. Otherwise F(G) = A × B with nontrivial normal subgroups A and B of G by (2). Obviously, σ|_{AK} is a class-preserving automorphism, so by minimality of G, σ|_{A} = conj(z) for some z ∈ Z(K). The same is true for B, and by Lemma 1 we may assume that σ|_{B} = id—observe that σ|_{K} = id still holds.

We shall show that for any b ∈ B, the restriction of σ to AC_{K}(b) is a classpreserving automorphism (and consequently an inner automorphism, by minimality of G). Take a ∈ A and k ∈ C_{K}(b) satisfying [a, k] = 1. Then [ab, k] = 1, and by Lemma 2, (abk)σ = (abk)^{x} for some x ∈ K. Equivalently, (ab)σ = (ab)^{x} and x ∈ C_{K}(b), so the criteria of Lemma 2 is satisfied.

Again using Lemma 2, we prove that the restriction of σ to F(G)Z(K)C_{K}(A) is a class-preserving automorphism. Take a ∈ A, b ∈ B and k ∈ Z(K)C_{K}(A) satisfying [ab, k] = 1. We have just proved that σ|_{AC_{K}(b)} = conj(g) for some g ∈ C_{K}(b). Now (abk)σ = (abk)^{g} as g, k ∈ C_{K}(b), and conj(z)|_{A} = σ|_{A} = conj(g)|_{A} implies g ∈ Z(K)C_{K}(A), so the criteria of Lemma 2 is satisfied. However, by (6), σ restricted to F(G)Z(K)C_{K}(A) is not an inner automorphism, so K = Z(K)C_{K}(A) by minimality of G. Of course the same is true for B, so K operates as an abelian group on F(G). Hence G is metabelian, contradicting (3).

(8) Let 1 ≠ m ∈ M = F(G) and k ∈ K with mσ = m^{k}. Then k ∉ F(K).

Proof. Otherwise we may assume that k ∈ O_{p}(K) by Lemma 1. Then k ∈ Z(K) by (5), and as M = ⟨m^{x} : x ∈ K⟩ by (7), it follows that σ|_{M} = conj(k)|_{M}, contradicting (6).

(9) M is an irreducible F(K)-module. We may identify G with a subgroup of an affine semi-linear group AΓ(F), with M = A(F), F(K) ≤ Γ_{0}(F) and K ≤ Γ_{0}(F) · Gal(F/E), where E = End_{K}(M) and [F : E] = [K : F(K)].

Proof. Consider M as F(K)-module. By Clifford’s Theorem and (7), M is a semisimple module, and K acts via conjugation on its homogenous components.

Let V be a homogenous component; we shall prove V = M. Assume that V ≠ M. As σ|_{M} is a F(K)-module automorphism, V σ = V, and using Lemma 2, it is easily verified that σ restricted to H := V N_{K}(V) is a class-preserving automorphism. By minimality of G, σ|_{H} = conj(k)|_{H} for some p-element k of K. Now F(K) ≤ N_{K}(V) and σ|_{K} = id imply that k ∈ C_{K}(F(K)) ≤ F(K), in contradiction to (8). Hence M is a homogeneous F(K)-module, and the Claim follows from Lemma 4.

We may assume that σ^{p} is an inner automorphism, and we fix z ∈ G such that σ^{p} = conj(z). Then z ∈ Z(K). Since σ|_{M} as well as conj(k)|_{M} for k ∈ Z(K) are contained in the multiplicative group of the finite field E = End_{K}(M), which is cyclic, it follows that Z(K) = ⟨z⟩. In this context, recall that Z(K) = O_{p}(K).

(10) G is a Frobenius group. There is a Sylow p-subgroup P = ⟨x⟩ with x^{p} = z ≠ 1.

Proof. By (9), K̄ = K/F(K) is cyclic, say K̄ = ⟨x̄⟩ for some x ∈ K. By (4), we may assume that x is a p-element. Identify G with a subgroup of AΓ(F) as above and write x = ϕ^{n} · c for some c ∈ Γ_{0}(F) = F^{×} and some power ϕ^{n} of the Frobenius automorphism ϕ. Note that Gal(F/E) = ⟨ϕ^{n}⟩, so if d = [F : E], then x^{d} = (ϕ^{n} · c)^{d} = N_{E}^{F}(c), the norm of c from F to E.

Assume that N_{E}^{F}(c) = 1. Then c^{−1} = v^{−1}(vϕ^{n}) for some v ∈ F^{×} by Hilbert’s Theorem 90. Equivalently v^{x} = v, if v is considered as an element of A(F) = F, so vσ = v^{k} for some k ∈ F(K), contradicting (8). Hence 1 ≠ x^{d} ∈ O_{p}(K).

P = ⟨z, x⟩ is a Sylow p-subgroup P of K; write P = ⟨s⟩ × ⟨t⟩ with s, t ∈ P. As P/⟨z⟩ is cyclic, we may assume that z = st^{l} for some l ∈ ℕ. We shall show that s = 1. Let N = O_{p′}(K); then K = N ⋊ P. Take any m ∈ M. Then mσ = m^{hs^{i}t^{j}} for some h ∈ N, i, j ∈ ℕ. It follows that m^{st^{l}} = m^{z} = mσ^{p} = m^{ks^{ip}t^{jp}} for some k ∈ N and m^{s} = m^{g} for some g ∈ H := N⟨s^{p}, t⟩. Application of Lemma 2 shows that σ induces a class-preserving automorphism on MH. If s ≠ 1, then MH < G, so σ induces an inner automorphism of MH by minimality of G, which is impossible by (6). Hence s = 1 and P is cyclic, P = ⟨x⟩.

The last two paragraphs show that G is a Frobenius group. Again by minimality of G, we may assume that mσ = m^{x} for some 1 ≠ m ∈ M. It follows that m^{z} = mσ^{p} = m^{x^{p}} and x^{p} = z.

(11) For all m ∈ M, there is k(m) ∈ F(K) such that mσ = m^{xk(m)}.

Proof. Let m ∈ M \{1}. By Lemma 1 and (10), mσ = m^{k^{−1}yk} for some y ∈ P and k ∈ F(K). It follows that m^{x^{p}} = m^{z} = mσ^{p} = m^{k^{−1}y^{p}k}, and, as G is a Frobenius group, x^{p} = z = k^{−1}y^{p}k. Hence (x^{−1}y)^{p} = 1 and x^{−1}y ∈ O_{p}(K), so mσ = m^{xk(m)} with k(m) = x^{−1}k^{−1}x(x^{−1}y)k ∈ F(K).

Now we are in a position to finish the proof of the theorem.

Identify G with a subgroup of AΓ(F) as in (9). By (10), [F : E] = p, so |F | = r^{pn} and |E| = r^{n} for some n ∈ ℕ. Note that p^{2}|r^{n} − 1 since O_{p}(K) ≠ 1 by (10) (see the paragraph following (9)).

We may assume that x = ϕ^{n} · c for some c ∈ Γ_{0}(F) = F^{×} (replacing both x and σ by a suitable power of them, if necessary).

Let ⟨ω⟩ be the Sylow p-subgroup of F^{×}, of order p^{a} (say). Then ⟨ω^{p}⟩ is the Sylow p-subgroup of E^{×} (see [12, VI 6.2(i)], or equation (1) in the proof of Theorem B).

Since ϕ^{n} is of order p, and we are only interested in the isomorphism type of G, we may assume that c ∈ ⟨ω⟩. Then d := N_{E}^{F}(c) = (ϕ · c)^{p} ∈ K and ⟨d⟩ = ⟨c^{p}⟩. As E = End_{K}(M), it follows that c ∈ ⟨ω^{p}⟩. Clearly c is not of order p (since x is not of order p). Hence we may assume that c = ω^{p^{b}} for some 1 ≤ b ≤ a − 2.

Let σ|_{M} be given by multiplication with μ ∈ E^{×}. Take any v ∈ M\{1} = F \{0}. Then vμ^{p} = vσ^{p} = v^{z} = v^{x^{p}} = vc^{p}, so μ and c (∈ F^{×}) differ only by an element of order p, μ = cλ (say). By (11), vμ = vσ = (vϕ^{n})cζ for some ζ ∈ F(K) ≤ F^{×}, and it follows that v^{−1}(vϕ^{n}) = λζ^{−1}. Note that λ ∈ O_{p}(K) since O_{p}(K) ≠ 1 by (10). Hence H = {v^{−1}(vϕ^{n}) | v ∈ F^{×}} =: ⟨η⟩ is a subgroup of K, and σ induces a class-preserving automorphism on MHP. By minimality of G, it follows that K = HP. Altogether, we have K = ⟨η^{p}, ϕ^{n} · ω^{p^{b}}⟩.

Note that v ↦ v^{−1}(vϕ^{n}) defines a homomorphism F^{×} → F^{×} whose image is E^{×}. Hence H is of order |F^{×}|/|E^{×}|, and G is isomorphic to the group G(p^{a−b}, r, n) defined in Theorem A′.

6. Proof of Theorem B

The proof is splitted into two parts, according to whether G is solvable or not.

Part 1: G is solvable. Let G be a minimal counterexample. Then G possesses a class-preserving, non-inner automorphism σ of order a power of p, for some prime p. Note that the class of groups described in the theorem is closed under subgroups and factor groups.

Assume that p ≠ 2. Then Sylow p-subgroups of G are cyclic, and it follows from Lemma 3(ii), (iii) that F(G) is not cyclic. Hence O_{2}(G) is not cyclic. By Lemma 3(i), (ii) and Theorem 1, O_{2}(G) is elementary abelian. As O_{2}(G) contains a cylic subgroup of index 2, it follows that O_{2}(G) ≅ C_{2} × C_{2}. Let O_{2′}(F(G)) = ⟨x⟩ and take 1 ≠ y ∈ O_{2}(G). By minimality of G we may assume that σ induces the identity on G/F(G). Since F(G) is abelian, Lemma 1 tells us that (xy)σ = (xy)^{g} for some p-element g ∈ G. Let α := σ|_{O_{2}(G)} and β := conj(g^{−1})|_{O_{2}(G)} as automorphisms of O_{2}(G). As F(G) is abelian and σ induces the identity on G/F(G), the automorphisms α and β commute, and so αβ is of order a power of p. On the other hand, αβ fixes the non-trivial element y, and the structure of Aut(O_{2}(G)) = GL(2, 2) = S_{3} implies that αβ is of order a power of 2. It follows αβ = id, and therefore σ|_{O_{2}(G)} = conj(g)|_{O_{2}(G)}, contradicting Lemma 3(iii).

The case p = 2 is more difficult to handle. We use the following description of G given by H. Zassenhaus.

Theorem 2 (H. Zassenhaus [20, Satz 6]). Let G be a finite solvable group, and assume that a Sylow 2-subgroup of G contains a cyclic subgroup of index 2. Then there is a normal subgroup N of G with cyclic Sylow 2-subgroup such that G/N is isomorphic to C_{2}, to the alternating group A_{4} of degree 4, or to the symmetric group S_{4} of degree 4.

Choose N ⊴ G as in the theorem. Note that N ≠ 1.

(1) O_{2′}(G) ≠ 1.

Proof. Assume that O_{2′}(G) = 1. Then F(N) is a (cyclic) 2-group, so Aut(F(N)) is a 2-group (see [9, I 13.19]), and it follows that N is a cyclic 2-group. Certainly G is not a 2-group, so there is N ◁ L ⊴ G such that L/N ≅ A_{4}. As Aut(N) is an abelian 2-group, it follows that N ≤ Z(L), and C_{2} ≅ N = Z(G) because of the special structure of the Sylow 2-subgroups (those of L are dihedral groups or generalized quaternion groups). By minimality of G we may assume that σ induces the identity on G/N. It follows that G has C_{2} as epimorphic image, and consequently G/N ≅ S_{4}. Let x be an element of order 8 in G. Then Z(G) = ⟨x^{4}⟩ and xσ = x^{5} since x ∉ L. Now {x, x^{−1}} is a conjugacy class in every Sylow 2subgroup of G which contains x, and x^{g} ∈ ⟨x⟩ for some element g ∈ G of order 3 implies x^{g} = x. Hence x is not conjugate to x^{5} (=xσ), a contradiction.

By (1), there is a Sylow subgroup Q of F(G) of odd order. By Lemma 3(i) and Theorem 1, Q is elementary abelian, with a complement K in G. As Q is contained in a cyclic Sylow subgroup of G, it follows that Q is a normal Sylow subgroup of G and of prime order. Let L := C_{K}(Q) ⊴ G. Then K/L is cyclic since it is isomorphic to a subgroup of the cyclic group Aut(Q). Let S be a Sylow 2-subgroup of K. Then L ∩ S is a Sylow 2-subgroup of L. By minimality of G we may assume that σ induces the identity on G/Q, and by Proposition 1 we may assume that kσ = k for all k ∈ K. The order of K/L is even; otherwise we would obtain σ = id by Lemma 1. It follows that the (cyclic) Sylow 2-subgroup of K/L is isomorphic to C_{2}, since it is an epimorphic image of a Sylow 2-subgroup of G. To summarize:

• Q = ⟨x⟩ is a normal Sylow subgroup of G of odd order;

• K is a complement to Q in G with kσ = k for all k ∈ K;

• L := C_{K}(Q) ⊴ G;

• S is a Sylow 2-subgroup of K, with L ∩ S a Sylow 2-subgroup of L and [S : L ∩ S] = 2.

(2) O_{2}(G) is not cyclic.

Proof. Otherwise F(G) is cyclic, contradicting Lemma 3(iii).

(3) L ∩ S is not cyclic.

Proof. Immediate from (2) since O_{2}(G) ≤ L ∩ S.

(4) There is no normal subgroup of G of order 2.

Proof. Assume that there is C_{2} ≅ M ⊴ G. Then by minimality of G, there is a 2-element g ∈ G such that σ induces on Ḡ = G/M the inner automorphism conj(ḡ). As sσ = s for all s ∈ S and M ≤ Z(S) ≤ S, we conclude g ∈ S and ḡ ∈ Z(S/Z(S)).

Assume that S is not a dihedral group of order 8. Then |S| ≥ 16 by (3), and we find that g is contained in each maximal subgroup of S, in particular in L ∩ S. We obtain the contradiction xσ ∈ xM ∩ Q = {x}.

Hence S is a dihedral group of order 8. By (3), L ∩ S = M × ⟨s⟩ for some s ∈ S. By Lemma 1, (sx)σ = (sx)^{k} for some 2-element k ∈ G. It follows that [s, k] = 1, so ⟨L ∩ S, k⟩ is an abelian Sylow 2-subgroup of G, contradicting (2) and Lemma 3(ii).

(5) G/N ≅ C̸_{2} and G/N ≅ A̸_{4}.

Proof. Assume the contrary. By (4), O_{2}(G) ∩ N = 1, and using (2) it follows that O_{2}(G) is isomorphic to the abelian Sylow 2-subgroup of G/N. Hence if Z is a Sylow 2-subgroup of N, then O_{2}(G)Z is an abelian Sylow 2-subgroup of G, contradicting (2) and Lemma 3(ii).

(6) G/N ≅ S̸_{4}.

Proof. By (4), O_{2}(G)∩N = 1, and using (2) it follows that Ḡ = G/O_{2}(G)N ≅ S_{3}. Since q ≤ N, there is an element g ∈ K of order 3. Then g ∈ L as G′ ≤ C_{G}(Q). By Lemma 1, (gx)σ = (gx)^{k} for some 2-element k ∈ K. It follows that [g, k] = 1 and xσ = x^{k}. The order of N is odd; otherwise [O_{2}(G), N] = 1 would imply that G contains an elementary abelian subgroup of order 8. Together with Q ≤ N, [O_{2}(G), N] = 1 and xσ = x^{k}, it follows that k ∉ O_{2}(G)N. But then Ḡ = ⟨ḡ, k̄⟩ is abelian, a contradiction.

Of course (5) and (6) contradict the choice of N, so we have shown that a minimal solvable counterexample cannot exist.

Part 2: G is not solvable. In this case, we can use the following description of M. Suzuki.

Theorem 3 (M. Suzuki [18, Theorems B, C]). Let G be a finite non-solvable group. Assume that Sylow subgroups of G of odd order are cyclic, and that a Sylow 2subgroup of G is either (i) a dihedral group, or (ii) a generalized quaternion group. Then there is G_{1} ⊴ G such that [G : G_{1}] ≤ 2 and G_{1} = Z × N, where all Sylow subgroups of Z are cyclic, and N is isomorphic to PSL(2, p) in case (i) and to SL(2, p) in case (ii), for some odd prime p.

Note that a group Z whose Sylow subgroups are all cyclic is necessarily metacyclic (see [9, IV 2.11]), so in the above situation, the group G/N is solvable.

For some prime p, let L = SL(2, p) and L̄ = PSL(2, p) = L/Z(L). (The Sylow subgroups of L and L̄ are described in [9, II 8.10]).

(1) Let p be an odd prime. Then Out(L) and Out(L̄) are of order 2, and Out_{c}(L) = 1, Out_{c}(L̄) = 1.

Proof. By [18, Lemma 2], Out(L) and Out(L̄) are of order 2. Let ω be a primitive element of 𝔽_{p}. By [18, pp. 688-9], a non-inner automorphism of L is defined by

θ : [ \begin{matrix}α & β \\ γ & δ \end{matrix} ] ↦ [ \begin{matrix}0 & −1 \\ ω & 0 \end{matrix} ] [ \begin{matrix}α & β \\ γ & δ \end{matrix} ] [ \begin{matrix}0 & −1 \\ ω & 0 \end{matrix} ]^{−1} = [ \begin{matrix}δ & −ω^{−1}γ \\ −ωβ & α \end{matrix} ] .

Let x = [ \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ]. Assume that there is g = [ \begin{matrix}α & β \\ γ & δ \end{matrix} ] ∈ L such that \overline{xθ} = ḡ^{−1}x̄ḡ. Then

[ \begin{matrix}1 & 0 \\ −ω & 1 \end{matrix} ] = ± [ \begin{matrix}1+δγ & δ^{2} \\ −γ^{2} & 1−δγ \end{matrix} ]. It follows that δ = 0, and the contradiction ω = γ^{2} as

p > 2. Hence θ induces an automorphism θ̄ of L̄ which is not class-preserving.

For short, we shall say that G belongs to D or Q if G is of type (i) or (ii) described in Theorem 3.

(2) Assume that G belongs to D or Q, and that there is N ⊴ G of index 2 with N = L or N = L̄. Then there is x ∈ G such that conj(x)|_{N} = θ or conj(x)|_{N} = θ̄, respectively.

Proof. Let S be a Sylow 2-subgroup of G, and take y ∈ S \ N. We assume that conj(y)|_{N} ∈ Inn(N) and have to reach a contradiction. There is a 2-element m ∈ N with conj(y)|_{S∩N} = conj(m)|_{S∩N}. Clearly m ∈ S ∩ N since S ∩ N is a Sylow 2-subgroup of N. It follows that S = ⟨S ∩ N, ym^{−1}⟩ and ym^{−1} ∈ Z(S), contradicting the structure of S.

(3) Let G and N be as in (2). If G belongs to D, then Out(G) = 1. If G belongs to Q, then Out(G) ≅ C_{2}, and there is a non-inner automorphism of G fixing N elementwise.

Proof. Let τ be a non-inner automorphism of G. By (2), we may assume that τ|_{N} = id (note that N is a characteristic subgroup). Let G = ⟨N, x⟩ and n ∈ N such that xτ = nx. Then n ∈ Z(N) since τ|_{N} = id, and as L̄ is a non-abelian simple group, G belongs to Q. On the other hand, if G belongs to Q, and Z(N) = ⟨n⟩, then τ|_{N} = id and xτ = nx defines a non-inner automorphism of G.

(4) Let G and N be as in (2), with G belonging to Q. Then Out_{c}(G) = 1.

Proof. By (3), there is a non-inner automorphism τ of G with τ|_{N} = id, and we have to show τ ∉ Aut_{c}(G). By (2), there is x ∈ G with conj(x)|_{N} = θ, and

xτ = zx with z = [ \begin{matrix}−1 & 0 \\ 0 & −1 \end{matrix} ].

We use a non-trivial fixed point in N under the operation of τ. Let ω be a primitive element of 𝔽_{p}. Then there are ξ, ζ ∈ 𝔽_{p} with ζ ≠ 0 and ξ^{2} + ωζ^{2} = 1. This is because 1 + ω ≠ 0, so if 1 + ω = λ^{2} for some λ ∈ 𝔽_{p}, we may choose ξ = ζ = λ^{−1}, and otherwise ω(1 + ω) = λ^{2} for some λ ∈ 𝔽_{p}, and we may choose ξ = ωλ^{−1} and ζ = λ^{−1}. Hence k = [ \begin{matrix}ξ & ζ \\ −ωζ & ξ \end{matrix} ] ∈ L = N, with ζ ≠ 0. It follows that kθ = k and [x, k] = 1.

Assume that τ ∈ Aut_{c}(G). Because of [x, k] = 1, there is g = [ \begin{matrix}α & β \\ γ & δ \end{matrix} ] ∈ N with

zkx = (kx)τ = (kx)^{g}. It follows that g^{k} = (gθ)z = [ \begin{matrix}−δ & ω^{−1}γ \\ ωβ & −α \end{matrix} ]. As conjugate matrices have the same trace, we get α + δ = 0 and

[ \begin{matrix}ξα−ωζβ & ζα+ξβ \\ ξγ+ωζα & ζγ−ξα \end{matrix} ] = gk = k(gθ)z = [ \begin{matrix}ξα+ωζβ & ω^{−1}ξγ−ζα \\ −ωζα+ωξβ & −ζγ−ξα \end{matrix} ] .

Comparing the entries in the main diagonal shows that β = γ = 0 (since ζ ≠ 0). Comparing the other entries now gives α = 0, so g is not invertible, a contradiction. Hence τ ∉ Aut_{c}(G).

Now we are in a position to finish the proof. Let G be a minimal non-solvable counterexample. Choose G_{1}, Z and N as in Theorem 3. Then [G : G_{1}] = 2 by (1), and Z ≠ 1 by (3) and (4). Let σ be a non-inner, class-preserving automorphism of G. By (1) and (2) we may assume that σ|_{N} = id. Take g ∈ G with G = ⟨G_{1}, g⟩. Assume that σ|_{Z} ∉ Aut_{c}(Z). Then yσ ∉ y^{Z} for some y ∈ Z. It follows that for each n ∈ N, there are z ∈ Z and m ∈ N such that (yn)σ = (yn)^{zmg} = y^{zg}n^{mg}, in particular n = nσ = n^{mg}. This says that conj(g)|_{N} ∈ Aut_{c}(N), contradicting (1) and (2). Hence σ|_{Z} ∈ Aut_{c}(Z) = Inn(Z), and we may assume that σ|_{G_{1}} = id. By (3) and (4), σ induces the identity on G/Z. Further, σ induces a class-preserving automorphism of the solvable group G/N, so by Part 1, σ = conj(z̄) or σ = conj(z̄ḡ) on Ḡ = G/N for some z ∈ Z. In the first case, z ∈ Z(Z) and gσ = g^{z}, and we get the contradiction σ = conj(z). In the second case, z = zσ = z^{zg} = z^{g} and gσ ∈ g^{zg}N ∩ gZ = {g^{zg}} = {g}, so σ = id.

Theorem B is proven.

7. Some Applications

7.1. Variations of the Zassenhaus conjecture. Let G be a finite group, and assume that ℤG = ℤH as augmented algebras. In [11, 1.1], the following problems were formulated.

1. The Isomorphism problem asks whether there exists an isomorphism ρ : G → H.

2. The p-version of the Zassenhaus conjecture asks whether in addition such a ρ = ρ_{p} can be chosen in such a way that its restriction to a Sylow p-subgroup P is given by conjugation with a unit in ℚG.

3. We shall say that the p-version of the Zassenhaus conjecture holds simultaneously for all p, if there exists an automorphism ρ : G → H such that its restriction to every Sylow subgroup S of G is given by conjugation with a unit in ℚG (depending in general on S).

4. The Zassenhaus conjecture asks whether G and H are conjugate in the units

of ℚG. We remark that instead of trying to prove the existence of some isomorphism G → H directly (with G from a certain class of groups, which is closed under direct products), it is often more convenient to prove a certain variation of the Zassenhaus conjecture “for automorphisms” which is strong enough to imply a positive solution of the Isomorphism problem. See [11] and [10] for details. Here we restrict ourselves to record the following corollaries.

From Corollary 5, and [11, Theorem 1.3], we get the following extension of [11, Corollary 1.4].

Corollary 7. Assume that G/F(G) is an extension of an abelian q-group by an abelian q′-group, for some prime q. Then for ℤG, the p-version of the Zassenhaus conjecture holds simultaneously for all p.

The next Corollary has been proved in [11, Corollary 5.2] under the additional assumption that Frobenius complements are metacyclic.

Note that Frobenius complements belong to the class of groups described in Theorem B (see [9, V 8.7]). Hence from Theorem B, and [11, Theorem 5.1], we get

Corollary 8. Let G be a Frobenius group. Then for ℤG, the p-version of the Zassenhaus conjecture holds simultaneously for all p.

7.2. Coleman automorphisms of A-groups. In the study of the so-called Normalizer Problem for integral group rings (see [17], and also [8] for recent results), automorphisms σ of a finite group G satisfying the following condition (C) naturally arise. (C) For any Sylow subgroup P of G, there is γ ∈ Inn(G) such that σ|_{P} = γ|_{P}. This follows from an observation due to D. Coleman (see [17, 9.1]), which leeds to interesting consequences. Therefore we shall call automorphisms σ satisfying (C) Coleman automorphisms.

M. Mazur considered in [14] a certain “Property W” of a finite group G, namely that Coleman automorphisms σ of G with σ ∈ Aut_{c}(G) and σ^{2} ∈ Inn(G) are inner automorphisms. In [14, p. 6270], it has been speculated that groups with abelian Sylow 2-subgroups have Property W. The following example shows that even for A-groups this is not the case.

Example 1. Let a = [ \begin{matrix}2 & 1 \\ 3 & 2 \end{matrix} ], b = [ \begin{matrix}3 & 0 \\ 0 & 2 \end{matrix} ] and c = [ \begin{matrix}4 & 0 \\ 0 & 1 \end{matrix} ] as elements of GL(2, 5). Their orders are 3, 4 and 2, respectively. Put Y = ⟨a, b, c⟩. Then ⟨a⟩ ⊴ Y with abelian complement ⟨b, c⟩ of order 8. Let Y act naturally on M = 𝔽_{5} ⊕ 𝔽_{5}, and let N = ⟨n⟩ be a cyclic group of order 3 on which Y acts as a group of automorphisms with H := C_{Y} (N) = ⟨a, b⟩. Form the corresponding semidirect product G = (M × N) ⋊ Y. Clearly, G is an A-group.

Note that MH is the A-group of order 300 with a class-preserving, non-inner automorphism (Section 3, first example).

An automorphism σ of G ist defined by xσ = x for all x ∈ MY and nσ = n^{−1}. Clearly, σ is a non-inner Coleman automorphism with σ^{2} ∈ Inn(G). (For the

Sylow 3-subgroup P = ⟨n, a⟩, we have σ|_{P} = conj(bc)|_{P}.)

We shall show that σ is a class-preserving automorphism. A typical element of G is of the form g = n^{i} ·m·y with i ∈ ℕ, m ∈ M and y ∈ Y. As bc = [ \begin{matrix}2 & 0 \\ 0 & 2 \end{matrix} ] ∈ Z(Y), g^{bc} = n^{i}σ · m^{bc} · y. If y acts fixed-point free on M, it follows that gσ = g^{bcm′} for some m′ ∈ M. Otherwise y = 1, or y is conjugated to c = [ \begin{matrix}4 & 0 \\ 0 & 1 \end{matrix} ] or b^{2}c = [ \begin{matrix}1 & 0 \\ 0 & 4 \end{matrix} ] by an element of ⟨a⟩. We already know that there is h ∈ ⟨a, b^{2}⟩b with m^{bc} = m^{h}, so if y = 1, then gσ = g^{bch^{−1}}. If y = c^{d} for some d ∈ ⟨a⟩, then conjugation with k := b^{d} = [ \begin{matrix}3 & 0 \\ 0 & 2 \end{matrix} ]^{d} squares the elements of C_{M}(y), and k commutes with n^{i}σ and y, so there is m′ ∈ [M, y] with gσ = g^{bck^{−1}m′}. If y = (b^{2}c)^{d} for some d ∈ ⟨a⟩, take k := (b^{3})^{d} = [ \begin{matrix}2 & 0 \\ 0 & 3 \end{matrix} ]^{d} and argue in the same way to conclude that g and gσ are conjugate.

Acknowledgements

I thank W. Kimmerle for suggesting me to work at such interesting questions. I would like to express thanks to the Lehrstuhl B für Mathematik at the RWTH Aachen,

where I am currently as a member of a “Graduiertenkolleg”, and where this paper was written. The ideas contained herein are an extension of my Master’s thesis.

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

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