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Local analysis of the normalizer problem. (English) Zbl 0987.16015
Let \(G\) be a finite group and let \(RG\) be the group ring of \(G\) over the commutative ring \(R\). In the paper under review the author studies various subgroups of the group \(\text{Out}(G)\) of outer automorphism of \(G\). An essential ingredient of the author’s spectacular counterexample to the isomorphism problem for integral group rings was the existence of a non-trivial element in the kernel \(\text{Out}_R(G)\) of \(\text{Out}(G)\to\text{Out}(RG)\) for \(R=\mathbb{Z}\). The author proves a variety of results related to this group in an extremely elegant way. We only cite some of them. Noting by \(A\) the ring of algebraic integers in the complex numbers, he reviews an earlier paper of K. W. Roggenkamp and the reviewer [J. Pure Appl. Algebra 103, No. 1, 91-99 (1995; Zbl 0835.16020)] where an example of a group with \(\text{Out}_A(G)\neq 1\) was given. The author simplifies the presentation which was not optimal there and modifies the above group to obtain a group \(G\) of order 3200 with \(\text{Out}_A(G)\neq 1\). Using this improved method, the author proves that \(\text{Out}_A(G)\) is Abelian, and any finite Abelian group can occur for metablian \(G\). Moreover, using a modification of an argument of Krempa, he proves that \(\text{Out}_A(G)\) is in the centre of the group of class preserving automorphisms. Finally, the author studies the case of groups with Abelian Sylow 2 subgroups. He produces a group \(G\) with all Sylow subgroups Abelian and \(\text{Out}_A(G)\neq 1=\text{Out}_\mathbb{Z}(G)\).

MSC:
16S34 Group rings
20C10 Integral representations of finite groups
20F28 Automorphism groups of groups
20F29 Representations of groups as automorphism groups of algebraic systems
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16W20 Automorphisms and endomorphisms
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