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

LOCAL ANALYSIS OF THE NORMALIZER PROBLEM

MARTIN HERTWECK

Abstract. For a finite group G, and a commutative ring R, the automorphisms of G inducing an inner automorphism of the group ring RG form a group Aut_{R}(G). Let Aut_{int}(G) = Aut_{A}(G), where A is the ring of all algebraic integers in ℂ. Testing some σ ∈ Aut(G) for membership of Aut_{int}(G) is a total issue for localization, and then Clifford theory provides a powerful tool. It is proved that Aut_{int}(G)/Inn(G) is an abelian group, and can indeed be any finite abelian group. It is an outstanding question whether Aut_{ℤ}(G) = Inn(G) if G has an abelian Sylow 2-subgroup. This is shown to be true in some special cases, but also a group G with abelian Sylow subgroups and Aut_{int}(G) ≠ Inn(G) is given.

1. Introduction

The normalizer of a finite group G in the units of its integral group ring ℤG is an interesting object. Its study includes the study of central units, which is already a very difficult and broad subject. Moreover, there is an apparently “small” quotient of the normalizer, naturally isomorphic to a certain subgroup of Out(G), which measures the extend to which there are “non-obvious” units normalizing G. It has been studied already in [8, 12, 14, 17, 19, 25].

The aim of this paper is to get a better understanding of when this quotient is non-trivial, since a close investigation of certain examples led to the construction of the first counterexample to the so called “isomorphism problem for integral group rings” [8,9], and that is where our interest stems from. More generally group rings RG are considered, where R is a G-adapted ring, that is, an integral domain of characteristic 0 in which no prime divisor of |G| is invertible. Typical examples of G-adapted rings are

– R = ℤ_{π}, the semilocalization of ℤ at a finite set π of primes containing the set π(G) of prime divisors of |G|, i.e. the intersection of the localizations ℤ_{(p)} = {a/b : a, b ∈ ℤ, (p, b) = 1}, with p ∈ π;

– R = O, a ring of integers in an algebraic number field.

Date: July 20, 2000.

2000 Mathematics Subject Classification. Primary 20E36, Secondary 16S34, 16U70.

Key words and phrases. integral group ring; normaliser problem; integral isomorphism problem.

The research was partially supported by the Deutsche Forschungsgemeinschaft.

One reason for this is that it has been conventional wisdom in the community of researchers in this area that ℤ_{π} is as good as ℤ (cf. [27, p. 267]), while at the same time it is much easier to work with the semilocal ring ℤ_{π}G.

To be more precise, the following notation is introduced. For a commutative ring R, let Aut_{R}(G) be the group of automorphisms of G which induce an inner automorphism of RG. The quotient Out_{R}(G) = Aut_{R}(G)/Inn(G)—changing ‘A’ to ‘O’ will always have this fixed meaning—has two natural interpretations:

– Out_{R}(G) is the kernel of the natural map Out(G) → Out(RG);

– Out_{R}(G) ≅ N_{U}(G)/G · C_{U}(G), where U = U(RG) is the group of units in

RG. In particular, Out_{R}(G) = 1 means that the units in RG normalizing G are the ‘obvious’ ones. Note that Aut_{R}(G) is always contained in Aut_{ℚ}(G) = Aut_{c}(G), the group of class-preserving automorphisms (cf. [12, Proposition 2.5]). Hence from now on, it is assumed that R is G-adapted, aiming for a nice blending of group theory and ring theory. A group G is said to have the normalizer property (NP) if

(NP) Out_{ℤ}(G) = 1, or, equivalently, N_{U(ℤG)}(G) = G · Z(U(ℤG)).

This may as well be understood as the normalizer problem, i.e. the problem to ‘determine’ in some sense the groups G with (NP), a subject of research initiated by the questions [12, 3.7 Question] and [28, Problem 43].

By the way, there is an analog of the normalizer problem for so called unitary subgroups of integral group rings, which is discussed in [16], but there, ‘counterexamples’ are easily found.

Finding groups which does not satisfy (NP) is a difficult task. In positive direction, it has been shown that (NP) holds for groups with a normal Sylow 2subgroup [12, 3.6 Theorem], and for groups whose generalized Fitting subgroup is a p-group [8, 4.2.4]. In both cases, the proof relies on the following result, which is essentially due to D. B. Coleman [3].

Lemma (Coleman). Let P be a p-subgroup of G, and let S be a commutative ring with pS ≠ S. Then N_{U(SG)}(P) = N_{G}(P) · C_{U(SG)}(P).

(In this form, it appears first in [26, Proposition 1.14]; see also [12, Proposition 2.3].) It means that a unit of the group ring SG normalizing P acts by conjugation on P like a group element g ∈ G.

The automorphisms of G whose restriction to any Sylow subgroup equals the restriction of some inner automorphism of G form a characteristic subgroup Aut_{Col}(G) of G. Coleman’s result shows that Aut_{R}(G) ≤ Aut_{Col}(G) (recall that R is Gadapted), which may justify the notation (occasionally, automorphisms of Aut_{Col}(G) are called Coleman automorphisms, for short). This important fact is recorded explicitely:

(∗) Aut_{R}(G) ≤ Aut_{c}(G) ∩ Aut_{Col}(G).

K. W. Roggenkamp and A. Zimmermann [25] realized that the question whether Out_{R}(G) is trivial or not can be attacked by using some kind of a general theory (an improved and simplified version is given by Proposition 2.6). They furnished examples of finite, three-step abelian groups G with Out_{ℤ_{π(G)}}(G) ≅ C_{2}, and hence also with Out_{O}(G) ≅ C_{2} for some suitable chosen ring O of integers in an algebraic number field (cf. Theorem 2.1).

This result indicates that there ‘should be’ groups which do not satisfy (NP). Nevertheless it took some more time until it was shown that Out_{ℤ}(G) = 1 for their groups G, and the following was proved [8, 4.5, Theorem A], [9]: there are finite groups G with Out_{ℤ}(G) ≠ 1. One example is a metabelian group of order 2^{25} · 97^{2}, with normal Sylow 97-subgroup.

It should be remarked that if Out_{ℤ}(G) ≠ 1, then G is necessarily of even order. This follows from the following result of J. Krempa (a proof is given in [12, 3.2 Theorem]), since each prime divisor of the order of Aut_{c}(G) divides |G| [11, I 4 Aufgabe 12].

Lemma (Krempa). Out_{ℤ}(G) is an elementary abelian 2-group.

Krempa’s result has recently been generalized by M. Mazur [19]. In particular, if O is a ring of integers in a number field K such that the complex conjugation is central in the Galois group of the normal closure of K, then Out_{O}(G) is again an elementary abelian 2-group.

An old problem, posed first by G. Higman in 1940, is the “isomorphism problem for integral group rings”. Given finite groups X and Y, is it true that ℤX ≅ ℤY implies X ≅ Y? Though exciting results in positive direction have been achieved, it is now known that in general the question has a negative answer: a counterexample has been given in [8, 9]. It is not intended to go into details, but a few words may be in order. In this work, non-isomorphic groups X and Y of order 2^{21} · 97^{28} are constructed which have isomorphic group rings over ℤ. Though the main idea for the construction of X is based on the pioneering feat of K. W. Roggenkamp and L. L. Scott (cf. [15, 22, 27]), a substantial part of the construction involves a group G < X with Out_{ℤ}(G) ≠ 1. It follows that counterexamples which are constructed in that way must be necessarily of even order, by Krempa’s result. Thus the isomorphism problem for finite groups of odd order is still open.

It should be remarked that to that time, it has been known that a group G with Out_{R}(G) ≠ 1 gives rise to non-isomorphic (infinite polycyclic) groups X = G × ℤ and Y with RX ≅ RY. This observation of M. Mazur [18] has been refined in [8,9] insomuch that it is also applicable to finite groups (but then, one has to deal with N_{U(RG)}(G) in more detail).

The isomorphism problem may be formulated just as well for group rings OX, with O being the ring of integers in an algebraic number field, and then one may ask for counterexamples. In this context, the subgroup Aut_{int}(G) ≤ Aut(G), consisting of those σ ∈ Aut(G) which induce an inner automorphism of some group ring OX, with O as above, should play an important role.

In Section 3, it is proved that Out_{int}(G) is contained in the center of Out_{c}(G). In particular, Out_{int}(G) is an abelian group, and examples are given showing that Out_{int}(G) can be indeed any finite abelian group, in contrast to Krempa’s result. Hence further investigations may very well lead to counterexamples to the isomorphism problem for OG, with G finite of odd order.

In Section 2, it is first recalled that Out_{int}(G) is the intersection of all Out_{ℤ_{(p)}}(G), with p ranging over the prime divisors of |G|. This enables one to use Clifford theory in the study of Out_{int}(G), and a version for automorphisms is established (extending [25, Proposition 1]), which will be applied to produce various groups with Out_{int}(G) ≠ 1.

To the best of our knowledge, for any known class of groups with (NP) this property can be verified entirely group-theoretical, i.e. by showing that in (∗), the group on the right-hand side is a 2′-group. An exception might be the class of groups with abelian Sylow 2-subgroups (for which it is not known whether (NP) holds), which will be examined in Section 4.

It seems that for groups G with Out_{ℤ}(G) ≠ 1, the structure of the Sylow 2subgroups is the most sensitive towards modifications. In [19, p. 176], M. Mazur conjectured, with frankly very little supporting evidence, that

(Ab) Out_{ℤ}(G) = 1 if G has abelian Sylow 2-subgroups.

One reason for that might be seen in Lemma 4.1, which tells us that for a counterexample G of minimal order (if existing), O_{2}(G) = 1, so a non-inner group automorphism σ ∈ Aut_{ℤ}(G) cannot be a central group automorphism, i.e. cannot induce the identity on G/Z(G). However, in all so far known examples, Out_{ℤ}(G) were covered by the group of central group automorphisms; a first example in a completely different direction is Example 4.12, where a group G with abelian Sylow subgroups and Out_{int}(G) ≠ 1, but Out_{ℤ}(G) = 1, is given. Nevertheless, (Ab) still remains a challenge.

Mazur proved (Ab) when Sylow 2-subgroups are of order 2. In Section 4, this result is extended to groups G with Sylow 2-subgroups being cyclic, or abelian of exponent ≤ 4 with normal 2-complement.

In this context, note that according to [19, p. 176], K. W. Roggenkamp and Z. Marciniak produced a preprint where it is proved that (Ab) holds if in addition G is metabelian. However, this is a special case of what has been proved in [7,10]: If G is a metabelian group with abelian Sylow p-subgroups, then Out_{c}(G) is a p′-group.

The notation used 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}. Recall that the following characteristic subgroups of Aut(G) have been introduced.

– Aut_{c}(G) is the group of class-preserving automorphisms;

– Aut_{Col}(G) consists of those α ∈ Aut(G) such that for any Sylow subgroup P of G, there is γ ∈ Inn(G) with α|_{P} = γ|_{P} (‘Coleman automorphisms’ for short);

– Aut_{R}(G) consists of those α ∈ Aut(G) which induce an inner automorphism of the group ring RG;

– Aut_{int}(G) consists of those α ∈ Aut(G) which induce an inner automorphism

of OG, for some ring O of algebraic integers in an algebraic number field. Also, the corresponding ‘outer’ automorphism groups have been introduced.

2. Some theoretical background

Local-global principle for class-preserving automorphisms. Let G be a finite group, α ∈ Aut(G), and R an integral domain of characteristic 0. Then RG becomes an R(G × G)-module, denoted by _{1}RG_{α}, by letting m · (x, y) = x^{−1}m(yα) for all m ∈ RG and x, y ∈ G (and linear extension of this operation). Note that 1RG_{α} ≅ _{1}RG_{1} if and only if α induces an inner automorphism of RG. This interpretation of an automorphism as an invertible bimodule allows us to prove a local-global principle for class-preserving group automorphisms (see [25], and [6] for more details).

Theorem 2.1. For any finite group, there is a ring of integers O in an algebraic number field such that

Out_{int}(G) = ⋂_{p | |G|} Out_{ℤ_{(p)}}(G) = Out_{ℤ_{π(G)}}(G) = Out_{O}(G).

Proof. Let α ∈ Aut(G). Then the bimodules _{1}ℤG_{α} and _{1}ℤG_{1} are in the same genus if and only if α belongs to ⋂_{p} Aut_{ℤ_{(p)}}(G), or, equivalently, to Aut_{ℤ_{π(G)}}(G) (cf. also Remark 2.2). By a theorem of H. Jacobinski [13, Satz 7] (see also [4, 51.33]) there is some ring O of algebraic integers such that _{1}ℤG_{α} and _{1}ℤG_{1} are in the same genus if and only if _{1}OG_{α} and _{1}OG_{1} are isomorphic. It remains to show that Aut_{int}(G) is contained in Aut_{ℤ_{(p)}}(G) for all p, which follows from a generalization of the Noether-Deuring Theorem [21].

Remark 2.2. Let α ∈ Aut(G) such that there are units u_{p} in ℤ_{(p)}G with α = conj(u_{p}), for all p ∈ π = π(G). Without lost of generality, each u_{p} is contained in ℤG. Let m = ∏_{p∈π} p. Then u = ∑_{p∈π} \frac{m}{p}u_{p} is a unit in ℤ_{π}G with α = conj(u). Indeed, rad(ℤ_{π})G ⊆ rad(ℤ_{π}G) and ℤ_{π}G/rad(ℤ_{π})G ≅ ⨁_{p∈π} 𝔽_{p}G, so u is a unit modulo the radical and hence a unit in ℤ_{π}G (see [4, 5.10]).

Remark 2.3. K. W. Roggenkamp pointed out where the obstruction for getting globally central automorphisms from local data lies [23, p. 82]. Let R be a Dedekind ring of characteristic 0, and let Cl_{RG}(Z(RG)) be the subgroup of the locally free class group Cl(Z(RG)) consisting of those isomorphism classes of invertible ideals 𝔞 in Z(RG) so that 𝔞RG is a principal ideal in RG. Then Fröhlich’s localization sequence (cf. [4, 55.25, 55.26]) can be extended to the following diagram with exact rows.

[figure: 1; Cl(Z(RG)); Picent(RG); ∏_{P ∈max(R)}Picent(R_{P} G); 1]

[figure: 1; Cl_{RG}(Z(RG)); Outcent(RG); λ; ∏_{P ∈max(R)}Outcent(R_{P} G); ≅]

In particular, the image of the natural map Out_{int}(G) → Outcent(RG) is a subgroup of Cl_{RG}(Z(RG)).

It is an open problem (see [23, IX 1.13]) whether λ is surjective, i.e. whether for any M in Picent(RG) there is an invertible bimodule in the same genus as M which is RG-free from the left (say). If R is a semilocal ring, then λ is an isomorphism (see [4, 55.26, 55.16]).

Clifford theory for automorphisms. Let N be a normal subgroup of a finite group G, and R a commutative ring. A classical result of Clifford describes the blocks of RG which do not have N in its kernel in terms of blocks of inertia groups of central idempotents of RN (see [2], [20, 6 §1 Lemma 1.7], and [23, Part 1, XIII] for a more thorough treatment of integral Clifford theory). This will yield criteria for when certain automorphisms of G induce inner automorphisms of these blocks.

Notation 2.4. For a finite group K, the following notation for idempotents is used:

ϵ_{K} = \frac{1}{|K|} ∑_{k∈K}k and η_{K} = 1 − ϵ_{K}.

Moreover, the following notation is fixed.

• G a finite group;

• N a normal subgroup of G;

• R a commutative ring;

• η an idempotent of RN, central in RG, with central primitive decomposition η = e_{1} + … + e_{s} in RN;

• E a set of representatives of the orbits on {e_{1}, …, e_{s}} under the operation of G;

• T(e) inertia group {g ∈ G : e^{g} = e}, for all e ∈ E.

Note that it is not required that the inertia groups T(e) are normal subgroups of G. Let us recall the following well-known theorem.

Theorem 2.5. With notation as above,

η · RG = ∏_{e∈E} ( ⨁_{T(e)g} ⨁_{T(e)h} g^{−1} · e RT(e) e · h) ≅ ∏_{e∈E} Mat_{|G:T(e)|}(e RT(e) e).

Proof. Clearly, η · RG is the sum of its R-submodules

A_{e,g,k,h} = g^{−1}eg · RT(e) · k · h^{−1}eh,

where e ranges over E and the g, h, k are taken from a set of right coset representatives X of T(e) in G. Assume that s = g^{−1}eg · t · k · h^{−1}eh ≠ 0 for some t ∈ T(e). Then t′ = gtkh^{−1} ∈ T(e), and s = g^{−1}et′eh. Conversely, given t′ ∈ T(e), then T(e)g^{−1}t′h = T(e)k for some k ∈ X, and it follows that t = g^{−1}t′hk^{−1} ∈ T(e), so g^{−1}et′eh ∈ A_{e,g,k,h}. Hence ∑_{k∈X} A_{e,g,k,h} = g^{−1} · e RT(e) e · h, and it follows that

η · RG = ∏_{e∈E} ( ⨁_{T(e)g} ⨁_{T(e)h} g^{−1} · e RT(e) e · h).

Fix some e ∈ E, and let {g_{1}, …, g_{n}} be a set of right coset representatives of T(e) in G. Then the direct factor of η · RG belonging to e can be written conveniently as

M_{e} = [ \begin{matrix}g_{1}^{−1} · e RT(e) e · g_{1} & ⋯ & g_{1}^{−1} · e RT(e) e · g_{n} \\ ⋮ & & ⋮ \\ g_{n}^{−1} · e RT(e) e · g_{1} & ⋯ & g_{n}^{−1} · e RT(e) e · g_{n} \end{matrix} ] ,

whence is isomorphic to the matrix ring Mat_{|G:T(e)|}(e RT(e) e).

The theorem allows a proof of the following proposition, which provides a useful criteria of when certain automorphisms of G lie in Aut_{R}(G).

Proposition 2.6. Assume that there is σ ∈ Aut(G) such that for all e ∈ E, the automorphism σ stabilizes e and leaves every right coset of T(e) in G invariant. Then the following are equivalent.

(i) σ induces an inner automorphism of eRT(e)e, say conjugation with u_{e}, for all e ∈ E;

(ii) σ induces an inner automorphism of η · RG. In either case, σ induces an inner automorphism of η · RG, given by conjugation with

∑_{e∈E} ∑_{T(e)g} e^{g} · g^{−1}u_{e}(gσ) ∈ η · RG.

Proof. Let e ∈ E, and M_{e} the matrix ring defined above. If (i) holds, then the diagonal matrix

D = [ \begin{matrix}g_{1}^{−1} · u_{e} · g_{1}σ \\ & g_{2}^{−1} · u_{e} · g_{2}σ & & 0 \\ 0 & & ... \\ & & & g_{n}^{−1} · u_{e} · g_{n}σ \end{matrix} ]

lies in M_{e}, and σ is given on M_{e} by conjugation with D, so (ii) holds.

Note that σ fixes g_{i}^{−1} · e RT(e) e · g_{j}, for all i, j. Hence if (ii) holds, then σ is given on M_{e} by conjugation with a diagonal matrix, whose (1, 1)-entry is of the form g_{1}^{−1} ·u_{e} ·g_{1}σ, where u_{e} is a unit in eRT(e)e such that σ is given on eRT(e)e by conjugation with u_{e}, so (i) holds. The additional remark is just a reinterpretation of the matrix D.

Remark 2.7. In particular, the proposition provides a simple method for constructing class-preserving automorphisms of groups. Namely, if σ ∈ Aut(G) stabilizes N, induces an inner automorphism of G/N and satisfies condition (i) with η = η_{N} (for example, if σ induces inner automorphisms of all inertia groups), then σ ∈ Aut_{R}(G).

A precursor of the proposition is [25, Proposition 1], which has been used by Roggenkamp and Zimmermann to produce for the first time groups G with Out_{int}(G) ≠ 1.

The following proposition is essentially a special case of Proposition 2.6 and will be applied in Example 4.12.

Proposition 2.8. Assume that the finite group G is a semi-direct product V ⋊ H with V an elementary abelian p-group, for some prime p. Let σ ∈ Aut(G) be defined by hσ = h for all h ∈ H, and vσ = v^{m} for all v ∈ V and some fixed m ∈ ℕ. If U is a subgroup of index p in V, let N_{U} = N_{G}(U)/U, and denote by σ_{U} the automorphism of N_{U} induced by σ. Let R = ℤ[p^{−1}]. Then σ ∈ Aut_{R}(G) if and only if σ_{U} ∈ Aut_{R}(N_{U}) for all U.

Proof. Let U_{1}, …, U_{s} be the subgroups of index p in V, and let e_{i} = η_{V} ϵ_{U_{i}}. Then {e_{1}, …, e_{s}} is a complete set of orthogonal primitive central idempotents in η_{V} RV, with T(e_{i}) = N_{G}(U_{i}). Let σ ∈ Aut(G). By Proposition 2.6, σ ∈ Aut_{R}(G) if and only if σ induces an inner automorphism of e_{i} RT(e_{i}) e_{i}, for all i. Fix some e = e_{i}, and let U = U_{i}. Since σ induces the identity on ϵ_{V} RG = RG/V, and e ⊕ ϵ_{V} = ϵ_{U}, it follows that σ induces an inner automorphism of ϵ_{U} RT(e) = RN_{U} if and only if σ induces an inner automorphism of e RT(e) e. This proves the proposition.

With this knowledge at hand, it is not difficult to find groups G with Out_{int}(G) ≠ 1. To the end of this section, a relatively small example is given, which may be compared with the group of order 2^{7}·p_{1}^{2}·p_{2}^{2} of derived length 3 given by Roggenkamp and Zimmermann [25].

Example 2.9. A supersolvable metabelian group G of order 2^{7} · 5^{2} = 3200 is constructed with Out_{int}(G) ≠ 1. Let P = ⟨w : w^{8}⟩⋊(⟨b : b^{2}⟩×⟨c : c^{4}⟩)⋊⟨s : s^{2}⟩ with relations w^{b} = w^{−1}, w^{c} = w^{5}, [w, s] = [b, s] = 1 and c^{s} = w^{4}c. The group G has normal subgroups M and N of order 5 and is a semi-direct product G = (M ×N)⋊P with C_{P} (M) = ⟨w, b, s⟩ and C_{P} (N) = ⟨w, b, c^{2}s⟩. An automorphism ϕ ∈ Aut(G) is defined by ϕ|_{MN} = id|_{MN} and xϕ = x^{s} for all x ∈ P, so that ϕ restricted to MP, NP is given by conjugation with s, c^{2}s respectively. Assume that there is g ∈ G with ϕ = conj(g). Then it follows that g ∈ P, and g ∈ Z(P)s ∩ C_{P} (MN) = ⟨w^{4}, c^{2}⟩s ∩ ⟨w, b⟩ = ∅, a contradiction. Hence ϕ ∉ Inn(G).

Next, it is shown that ϕ ∈ Aut_{int}(G). Let R = ℤ_{(2)}[ζ_{5}], where ζ_{5} = exp(2πi/5). Then Proposition 2.6 may be applied with the normal subgroup N and η = η_{N} to get ϕ ∈ Aut_{R}(G) (since each inertia group equals MNC_{P} (N)). Interpreting ϕ as invertible bimodule, it follows from the Noether-Deuring theorem [21] that ϕ ∈ Aut_{ℤ_{(2)}}(G). Since ϕ is given by conjugation with the unit u = ϵ_{⟨w^{4}⟩}+η_{⟨w^{4}⟩}(w+w^{−1}) in ℤ[\frac{1}{2}]G, it follows by Theorem 2.1 that ϕ ∈ Aut_{int}(G).

It should be pointed out that the automorphism ϕ is a central group automorphism, that is, ϕ induces the identity on G/Z(G). In Example 4.12, an automorphism σ ∈ Aut_{int}(G) is presented which is not the product of an inner group automorphism and a central group automorphism.

3. Structure of Out_{int}(G)

A surprisingly simple proof for the following result is given.

Proposition 3.1. The group Out_{int}(G) is contained in the center of Out_{c}(G). In particular, Out_{int}(G) is an abelian group.

Proof. For any commutative ring R, let ∗ be the usual anti-involution of RG associated with the group basis G. Then for any u ∈ N_{U(RG)}(G) and g ∈ G,

guu^{∗} = u(u^{−1}gu)u^{∗} = u(u^{−1}g^{−1}u)^{∗}u^{∗} = uu^{∗}gu^{−∗}u^{∗} = uu^{∗}g,

showing that uu^{∗} ∈ Z(RG). Therefore, for any u, v ∈ N_{U(RG)}(G),

[u, v][u, v]^{∗} = u^{−1}v^{−1}(u(vv^{∗}))u^{∗}v^{−∗}u^{−∗}
= u^{−1}(v^{∗}(uu^{∗}))v^{−∗}u^{−∗}  (vv^{∗} commutes with u)
= u^{∗}v^{∗}v^{−∗}u^{−∗}  (uu^{∗} commutes with v^{∗})
= 1.

Let ϕ ∈ Aut_{c}(G) and α ∈ Aut_{int}(G); it has to shown that [ϕ, α] ∈ Inn(G). There is u ∈ U(ℚG) with ϕ = conj(u) and v ∈ U(OG) with α = conj(v) (for some ring O of algebraic integers). It follows that

[u, v] = u^{−1}\underbrace{(v^{−1}uv)}_{∈ℚG} = \underbrace{(u^{−1}v^{−1}u)}_{∈OG}v ∈ ℚG ∩ OG = ℤG.

As [u, v][u, v]^{∗} = 1, it follows that [u, v] = ±g for some g ∈ G (just look at the 1-coefficient of [u, v][u, v]^{∗} – this argument goes back to S. D. Berman). Hence [ϕ, α] = conj(g) ∈ Inn(G).

The picture would not be complete without examples showing that indeed any abelian group occurs as some Out_{int}(G). The groups given in the following proposition have a similar structure as the groups given by E. Dade in [5, § 2].

Proposition 3.2. Let q be a natural power of a prime p. Then there exists a finite metabelian group G such that Out_{int}(G) is cyclic of order q.

Proof. Let S = ⟨ν, τ, σ : ν^{q} = τ^{q} = σ^{q^{2}} = 1, [ν, τ] = [ν, σ] = 1, τ^{σ} = ντ⟩, a group of order q^{4}. Let K_{1} = ⟨ν, σ^{q}τ^{−1}⟩ and K_{2} = ⟨ν, τ⟩, both normal subgroups of S with S/K_{1} and S/K_{2} cyclic of order q^{2}, generated by the image of σ.

Choose primes p_{1} ≠ p_{2} such that pq does not divide p_{i} − 1. Put r = q/p. For i = 1, 2, let M_{i} be an indecomposable faithful 𝔽_{p_{i}}⟨σ^{r}⟩-module of finite dimension, and let P_{i} be the direct sum of r copies of M_{i}. Define a (faithful) operation of ⟨σ⟩ on P_{i} by letting (a_{1}, a_{2}, …, a_{r−1}, a_{r})^{σ} = (a_{r}σ^{r}, a_{1}, a_{2}, …, a_{r−1}), for all a_{j} ∈ M_{i}. Let R_{i} = ℤ[p_{i}^{−1}], and note that each central primitive idempotent of η_{P_{i}} · R_{i}P_{i} is of the form η_{P_{i}}ϵ_{U} for some maximal subgroup U of P_{i} (recall Notation 2.4 for idempotents). Since the module P_{i}|_{⟨σ^{r}⟩} decomposes into a direct sum of copies of M_{i}, it follows from the Krull-Schmidt theorem that σ^{r} operates fixed-point free on the set of central primitive idempotents belonging to η_{P_{i}} · R_{i}P_{i}.

Define the group G to be the semi-direct product (P_{1}×P_{2})⋊S with C_{S}(P_{i}) = K_{i} and the given action of σ.

Let e_{i} be a central primitive idempotent of η_{P_{i}} · R_{i}P_{i}. Then for the inertia group T(e_{i}) of e_{i} in G, it follows that [G, G] ≤ P_{1}P_{2}⟨ν⟩ ≤ T(e_{i}) ≤ P_{1}P_{2}⟨ν, τ, σ^{q}⟩.

Let β ∈ Aut_{int}(G). Seeking for a ‘canonical’ coset representative of β in Inn(G) · β, choose g_{i} ∈ P_{i} with C_{S}(g_{i}) = K_{i} (this can be done since S/K_{i} is cyclic). Then, as β ∈ Aut_{c}(G), the automorphism β can be altered by an inner group automorphism so that g_{i}β = g_{i}. By Coleman’s Lemma, there are s_{i} ∈ S with σ|_{P_{i}} = conj(s_{i})|_{P_{i}}. Then g_{i} = g_{i}β = g_{i}^{s_{i}}, and it follows that s_{i} ∈ K_{i}. Hence β fixes P_{1}P_{2} element-wise. Additionally, by Sylow’s theorem, β can be altered by an inner group automorphism so that β fixes S; then there is x ∈ S with β|_{S} = conj(x)|_{S}, again by Coleman’s Lemma.

Clearly e_{i}β = e_{i}, and β induces the identity on G/T(e_{i}). Hence β induces an inner automorphism of e_{i}R_{i}T(e_{i})e_{i}, by Proposition 2.6. Note that T(e_{2}) = P_{1}P_{2}H with K_{2} ≤ H ≤ S, and that β induces an automorphism of the quotient e_{2}R_{2}(P_{2}H)e_{2} of e_{2}R_{2}T(e_{2})e_{2}. Since τ ∈ Z(P_{2}H), it follows that τβ = τ and x ∈ ⟨ν, τ, σ^{q}⟩ = Z(S)⟨τ⟩. Thus without lost of generality, x is a power of τ.

Now define β ∈ Aut(G) by gβ = g for g ∈ P_{1}P_{2}, and β|_{S} = conj(τ)|_{S}. Then β induces the identity on the inertia group T(e_{i}), and on the abelian quotient G/T(e_{i}), so β ∈ Aut_{int}(G) by Proposition 2.6 and Theorem 2.1. Since Z(S)τ^{r} ∩ C_{P_{1}P_{2}}(S) = ⟨ν, σ^{q}⟩τ^{r} ∩ ⟨ν⟩ = ∅, it follows that the image of β in Out(G) has order q. This completes the proof of the proposition.

Corollary 3.3. For any finite abelian group A, there is a finite metabelian group G with Out_{int}(G) ≅ A.

Proof. Follows from Proposition 3.2 and Remark 4.3.1.

Corollary 3.4. For any finite abelian group A, there is a finite metabelian group G such that Cl_{ℤG}(Z(ℤG)) contains a copy of A (cf. Remark 2.3).

Proof. Follows from Proposition 3.2 and Krempa’s lemma.

4. Groups with abelian Sylow 2-subgroups

Let G be a finite group. If G has an abelian Sylow 2-subgroup P, it is conjectured that Out_{ℤ}(G) = 1 [19, p. 176]. This is shown to be true if P is cyclic, or of exponent ≤ 4 with normal 2-complement. However, also an example is given showing that Out_{int}(G) can be non-trivial (but Out_{ℤ}(G) = 1 for this example).

The following simple but important observation explains why it is difficult to disprove (NP) for groups with abelian Sylow 2-subgroups: for a counterexample G of minimal order, O_{2}(G) = 1 holds, and thus σ ∈ Aut_{ℤ}(G) cannot be a central group automorphism.

Lemma 4.1. Assume that G has an abelian Sylow p-subgroup P, and that σ ∈ Aut(G) is of p-power order such that σ|_{P} = γ|_{P} for some γ ∈ Inn(G). If σ induces an inner automorphism of G/O_{p}(G), then σ ∈ Inn(G).

Proof. Without lost of generality, σ induces the identity on G/O_{p}(G). There is x ∈ G with σ|_{P} = conj(x)|_{P}, and without lost of generality, x is a p-element since the order of σ is a power of p, and P is fixed by σ. It follows that x ∈ P, and, as P is abelian, σ|_{P} = id|_{P}. Now a common 1-cohomology argument [11, I 16.18] shows that σ is given by conjugation with some g ∈ Z(P).

Next, some elementary facts about class-preserving and Coleman automorphisms are recorded which will be needed later.

Proposition 4.2. 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 [11, I.4 Aufgabe 12]). Let σ ∈ Aut_{Col}(G), and assume that σ has order r, with (r, |G|) = 1; then σ = id has to be shown. 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.

Remark 4.3. 1. If σ is a class-preserving or a Coleman automorphism of G, and N a normal subgroup of G, then σ fixes N, and induces a class-preserving or a Coleman automorphism of G/N, respectively. In particular, Aut_{c}(G×H) ≅ Aut_{c}(G) × Aut_{c}(H) for finite groups G and H (and similar for Aut_{Col}(−)).

2. If σ ∈ Aut(G) is of p-power order, 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 ‘modifying σ (by an inner automorphism)’.

Lemma 4.4. Let α ∈ Aut(G) be of p-power order, for some prime p. Assume 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 following proposition has been proved for G supersolvable of odd order [17, Theorem 5], and under additional assumptions in [17, Theorem 6].

Proposition 4.5. Let G be a solvable group of odd order, and let ϕ, ψ be automorphisms of G such that:

(i) ϕ and ψ are of 2-power order; (ii) ϕ · ψ = ψ · ϕ; (iii) for all x ∈ G, x is conjugated in G to xϕ or to xψ. Then ϕ = id or ψ = id.

Proof. The hypothesis of the proposition is equally satisfied if ϕ is replaced by ϕ^{2}, or ψ by ψ^{2}, so without lost of generality, ϕ^{2} = id and ψ^{2} = id. Since G is solvable, there is a non-trivial, abelian normal subgroup A of G with Aϕ = A and Aψ = A, and whose order is minimal subject to this condition. The automorphisms ϕ and ψ induce automorphisms ϕ̄ and ψ̄ of Ḡ = G/A. By induction on the order of G, assume that ψ̄ = id. Then A = C_{A}(ψ) × [A, ψ] (coprime action since G is of odd order, cf. [1, 24.6]), and C_{A}(ψ) is a normal subgroup of G fixed by ϕ. By minimality of A, either C_{A}(ψ) = A or C_{A}(ψ) = 1. In the first case, ψ = id since G is of odd order, so assume that C_{A}(ψ) = 1. That is, aψ = a^{−1} for all a ∈ A, whence a is conjugated to aϕ for all a ∈ A by (iii). Since A = C_{A}(ϕ) × [A, ϕ] and ϕ inverts the elements of [A, ϕ], it follows that C_{A}(ϕ) = A. Put W = C_{G}(ψ) ≤ G. As ψ operates on each coset Ax, x ∈ G, with at least one fixed point, G = A ⋊ W. Since ϕ and ψ commute, W ϕ = W. Take any g ∈ G. Then a^{g} = a^{g}ϕ = a^{gϕ} for all a ∈ A, so c = (gϕ)g^{−1} ∈ C_{G}(A). Write g = wa with w ∈ W and a ∈ A. Then c = (wϕ)a · a^{−1}w^{−1} = (wϕ)w^{−1} ∈ W. Fix some 1 ≠ b ∈ A. Assume there is h ∈ G with (bc)^{h} = (bc)ψ = b^{−1}c. As bc ∈ C_{G}(A), and G = AW, there is h ∈ W with this property. But then bb^{h} = cc^{−h} ∈ A ∩ W = 1 as c ∈ W, so h inverts b, a contradiction. Therefore, by (iii), bc and (bc)ϕ are conjugate in G. Now cϕ = c^{−1} by definition of c, so (bc)ϕ = bc^{−1}, and it follows that c̄ and c̄^{−1} are conjugate in Ḡ = G/A. Therefore c̄ = 1 and c ∈ A ∩ W = 1. This shows that gϕ = g, and, as g was arbitrary, ϕ = id.

In [17], the following property of a finite group has been considered.

Property W. The group Out_{c}(G) ∩ Out_{Col}(G) is of odd order.

In [17, Remarks, p. 6270], it is asked whether there exist results similar to Proposition 4.5 for the action of abelian 2-groups on groups of odd order, which would allow one to prove that groups with abelian Sylow 2-subgroups have property W. Example 4.11 below shows that this is not the case.

M. Mazur has proved that a finite group G with a Sylow 2-subgroup of order 2 has property W [17, Theorem 7]. Note that such a group has a normal 2-complement, by Burnside’s normal p-complement theorem (cf. [1, 39.1]). Proposition 4.5 allows us to prove the following generalizations.

Proposition 4.6. Let G be a finite group with a Sylow 2-subgroup of order 2. Then 2 does not divide the order of Out_{c}(G).

Proof. Let σ ∈ Aut_{c}(G), of order a power of 2; it has to be shown that σ ∈ Inn(G). Let w ∈ G be an element of order 2, and H a normal 2-complement. Without lost of generality, wσ = w. Then τ = σ · conj(w) ∈ Aut(G) is of 2-power order, and στ = τσ. The automorphisms σ, τ induce automorphisms ϕ, ψ of H, respectively. Let h ∈ H. Then hσ = h^{g} for some g ∈ G = H ∪ Hw, and it follows that h is conjugate in H to hϕ or hψ. Hence by Proposition 4.5 ϕ = id or ψ = id, which implies σ = id or σ = conj(w).

Proposition 4.7. Let G be a finite group with a cyclic Sylow 2-subgroup. Then G has Property W.

Proof. Take a minimal counterexample G, and σ ∈ Out_{c}(G) ∩ Out_{Col}(G), a noninner automorphism of order a power of 2. The proof proceeds in a number of steps.

(1) O_{2}(G) = 1.

This follows from Lemma 4.1 and the minimality of G.

(2) The Frattini subgroup Φ(G) of G is trivial, and the Fitting subgroup F(G) is the direct product of minimal normal subgroups of G.

Assume the contrary. By minimality of G, the automorphism σ can be modified by an inner automorphism so that it induces the identity on G/Φ(G). As Φ(G) is nilpotent (see [11, III 3.6]), it follows from (1) that Φ(G) is a 2′-group, yielding the contradiction σ = id (see [11, III 3.18]). The second statement follows from the first one (see [11, III 4.4, 4.5]).

(3) G has more than one minimal normal subgroup (the trivial subgroup is not considered as being ‘minimal’).

Assume the contrary, and let A be the (unique) minimal normal subgroup of G. By Burnside’s normal p-complement theorem (see [1, 39.1]), G has a normal 2complement, so G is solvable, by the Odd Order Theorem. Therefore A is an abelian p-group for some prime p, and F := F(G) = O_{p}(G). Hence σ can be altered by an inner automorphism to get σ|_{F} = id|_{F}. As C_{G}(F) ⊆ F (see [11, III 4.2]), it follows that σ induces the identity on G/F. As p ≠ 2 by (1), it follows that σ = id, a contradiction.

(4) Any 2-element of G operates trivial or fixed-point free on each minimal normal subgroup of G.

To see this, let A be a minimal normal subgroup of G, and modify σ by an inner automorphism so that σ induces the identity on G/A. Then C_{A}(σ) ⊴ G, so either A = C_{A}(σ) or A = [A, σ]. In the first case, σ = id by (1), so A = [A, σ]. By Lemma 4.4, there is a 2-element x ∈ G with σ|_{A} = conj(x)|_{A}. It follows that A = [A, x], and that C_{A}(y) ⊴ G for all y ∈ ⟨x⟩. Since A is a minimal normal subgroup and Sylow 2-subgroups of G are cyclic, the assertion follows.

(5) Let S be a Sylow 2-subgroup of G. Then C_{S}(A) ≠ 1 for each minimal normal subgroup A of G.

Assume there is a minimal normal subgroup A of G with C_{S}(A) = 1. By (3), there is a minimal normal subgroup B of G with A ∩ B = 1. Without lost of generality, σ induces the identity on G/B. As in (4) it follows that B = [B, σ], and clearly σ|_{A} = id_{A}. Take any a ∈ A \ {1} and b ∈ B \ {1}. There is x ∈ G with (ab)σ = a^{x}b^{x}, and it is easily seen (cf. Lemma 4.4) that x can be chosen to be 2-element. It follows that x ∈ C_{G}(a) and x ∉ C_{G}(b). So x ∈ C_{G}(A) by (4), and x ≠ 1, which implies that C_{S}(A) ≠ 1.

Now the proof of the proposition is easily completed. As S is cyclic, it follows from (2), (5) and [11, III 4.2] that 1 ≠ C_{S}(F(G)) ≤ F(G), contradicting (1).

Corollary 4.8. If G has cyclic Sylow 2-subgroups, then Out_{ℤ}(G) = 1. □

Proof. By Krempa’s result, Out_{ℤ}(G) is a 2-group, so the corollary follows from Coleman’s result and the above proposition.

Up to now, considerations were entirely group-theoretical. The proof of the following lemma indicates further possibilities to analyze the group Out_{ℤ}(G).

Lemma 4.9. Let G be a finite group with a normal 2-complement H. Let P be a Sylow 2-subgroup of G, and S ⊴ P such that P/S is abelian of exponent ≤ 4. Then for any σ ∈ Aut_{ℤ}(G), there is γ ∈ Inn(G) such that σγ induces a Coleman automorphism of HS.

Proof. By assumption, there is a unit u of ℤG with σ = conj(u), and it does no harm if u is assumed to be augmented (i.e. the ‘sum over the coefficients’ is 1). Let N = HS. By a well-known result of G. Higman, u maps to an element of G/N under the natural map ℤG → ℤG/N (see [28, Theorem (2.7)]). Let g_{1}, …, g_{r} be a set of coset representatives of N in G. Then u = ∑ y_{i}g_{i} for some y_{1}, …, y_{r} ∈ ℤN, and without lost of generality, y_{i} has augmentation 1 if i = 1 (and augmentation 0 otherwise). Note that G acts (from the right) on the coset Ng_{1} via ng_{1} · g := g^{−1}(ng_{1})(gσ) (for all n ∈ N, g ∈ G). As g^{−1}(y_{1}g_{1})(gσ) = y_{1}g_{1}, it follows that in y_{1}g_{1}, viewed as an integral linear combination of elements of Ng_{1}, the elements of an orbit all have the same coefficient. Let p be a prime, and P a Sylow p-subgroup. Since the augmentation of y_{1}g_{1} is not divisible by p, there must be a fixed-point ng_{1} (n ∈ N) in the support of y_{1}g_{1} under the operation of P. It follows that σ|_{P} = conj(ng_{1})|_{P}, and σ · conj(g_{1}^{−1}) induces a Coleman automorphism of N.

Corollary 4.10. Let G be a finite group with a normal 2-complement and an abelian Sylow 2-subgroup of exponent ≤ 4. Then Out_{ℤ}(G) = 1.

Proof. Let H be the normal 2-complement, and σ ∈ Aut_{ℤ}(G); then σ ∈ Inn(G) has to be shown. By Krempa’s result, it can be assumed that the order of σ is a power of 2. By Lemma 4.9, σ can be modified by an inner automorphism such that σ induces a Coleman automorphism of H, so σ|_{H} = id|_{H} by Proposition 4.2. But then σ = id.

Finally, some illuminating examples of groups G with abelian Sylow subgroups are given. They include some Frobenius groups, which belong to a certain family of subgroups of affine semi-linear groups having non-inner class-preserving automorphisms; see [10] for further details on these groups.

Example 4.11. Consider the following matrices in GL(2, 5):

A = [\begin{matrix}2 & 1 \\ 3 & 2 \end{matrix}] , B = [\begin{matrix}3 & 0 \\ 0 & 2 \end{matrix}] , T_{1} = [\begin{matrix}4 & 0 \\ 0 & 1 \end{matrix}] .

Let U = 𝔽_{5} ⊕ 𝔽_{5} be the underlying vector space on which these matrices act. It is easily shown that

• A, B and T_{1} are of order 3, 4 and 2, respectively, and A^{B} = A^{−1}, A^{T_{1}} = A^{−1}. The group H = ⟨A, B⟩ = ⟨A⟩ ⋊ ⟨B⟩ has order 12.

• The semi-direct product F_{1} = U ⋊ H, i.e.

F_{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} ⟩,

is a Frobenius group.

• Each of the 6 = (5^{2} − 1)/(5 − 1) matrices of order 4 in H has eigenvalues 2 and 3. In particular, for each 0 ≠ u ∈ U there is M ∈ H such that uM = 2u.

• The automorphism σ_{1} of F_{1}, defined by uσ_{1} = 2u and hσ_{1} = h for all u ∈ U, h ∈ H, is a non-inner class-preserving automorphism.

Note that σ_{1} ∉ Aut_{ℤ_{(5)}}(F_{1}), by Coleman’s result.

On the other hand, let R = ℤ[\frac{1}{5}]. Then σ_{1} ∈ Aut_{R}(F_{1}). To see this, let 0 ≠ u ∈ U, and recall that there is M ∈ H of order 4 with uM = 3u, and u′M = 2u′ for some u′ ∈ U with U = ⟨u, u′⟩. Then N := N_{F_{1}}(⟨u⟩) = U⟨M⟩, and σ_{1} induces on N̄ = N/⟨u⟩ the inner automorphism conj(M̄). It follows from Proposition 2.8 that σ_{1} induces an inner automorphism of ℤ[\frac{1}{5}]F_{1}.

Let N = ⟨n⟩ ≅ C_{3} and T = ⟨t⟩ ≅ C_{2}. Let t act on N by n^{t} = n^{−1}, and on F_{1} ≤ U ⋊ GL(2, 5) by conjugation with the matrix T_{1}. Form the corresponding semi-direct product G = (N×F_{1})⋊T, and note that G has abelian Sylow subgroups.

The automorphism σ_{1} extends to an automorphism σ of G with nσ = n and tσ = t. Note that σ ∈ Aut_{Col}(G). As before, one can show that σ induces an inner automorphism of ℤ[\frac{1}{5}]G. In particular, σ ∈ Aut_{c}(G), and it follows that Property W need not hold for a group with abelian Sylow subgroups.

Let ζ = exp(2πi/3), and R = ℤ_{(5)}[ζ]. For the idempotent e = \frac{1}{3}(1 + ζn + ζ^{2}n^{2}), it is easily seen that e RT(e) e = e R(N × F_{1}) e ≅ RF_{1}. Hence condition (i) of Proposition 2.6 does not hold, and σ ∉ Aut_{ℤ_{(5)}}(G).

Example 4.12. A group of order 2^{5} · 3 · 7 · 5^{2} · 13^{2} = 2839200, with all Sylow subgroups being abelian, shall be constructed such that

Out_{int}(G) ≠ 1 and Out_{ℤ}(G) = 1.

Let F_{1} and T_{1} be as in the previous example.

The following matrices are elements of GL(2, 13):

C = [\begin{matrix}5 & 7 \\ 9 & 5 \end{matrix}] , D = [\begin{matrix}8 & 0 \\ 0 & 5 \end{matrix}] , T_{2} = [\begin{matrix}12 & 0 \\ 0 & 1 \end{matrix}] .

Let V be the underlying vector space. One verifies the following.

• C, D and T_{2} are of order 7, 4 and 2, respectively, and C^{D} = C^{−1}, C^{T_{2}} = C^{−1}. The group K = ⟨C, D⟩ = ⟨C⟩ ⋊ ⟨D⟩ has order 28.

• The semi-direct product F_{2} = V ⋊ K is a Frobenius group.

• Each of the 14 = (13^{2} − 1)/(13 − 1) matrices of order 4 in K has eigenvalues 5 and 8. In particular, for each 0 ≠ v ∈ V there is M ∈ K such that vM = 5v.

• The automorphism σ_{2} of F_{2}, defined by vσ_{2} = 5v and kσ_{2} = k for all v ∈ V,

k ∈ K, is a non-inner class-preserving automorphism. The group G is a semi-direct product

G = (F_{1} × F_{2}) ⋊ ⟨t : t^{2}⟩,

where the involution t acts on F_{1} ≤ U ⋊ GL(2, 5) by conjugation with the matrix T_{1}, and on F_{2} ≤ U ⋊ GL(2, 13) by conjugation with T_{2}.

Let σ be the automorphism of G which fixes each element of V HK⟨t⟩, and acts on U by multiplication with 2. Clearly, σ is a non-inner automorphism.

Let 0 ≠ u ∈ U. Recall that there is M ∈ H of order 4 with uM = 3u, and u′M = 2u′ for some u′ ∈ U with U = ⟨u, u′⟩. If M = A^{−i}B^{j}A^{i} (such i, j ∈ ℕ exist), then N := N_{G}(⟨u⟩) = UV ⟨M, A^{−i}tA^{i}⟩K. Hence σ induces on N̄ = N/⟨u⟩ an inner automorphism, given by conjugation with M̄. It follows from Proposition 2.8 that σ induces an inner automorphism of ℤ[\frac{1}{5}]G.

Note that σ · conj(B^{−1}tD) fixes each element of UHK⟨t⟩, and acts on V by multiplication with 5. In the same way as before, it is verified that σ induces an inner automorphism of ℤ[\frac{1}{13}]G.

Hence σ ∈ Aut_{int}(G) by Theorem 2.1. It follows from Corollary 4.10 that Out_{ℤ}(G) = 1. Thus everything is proved.

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