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Coleman automorphisms of finite groups. (English) Zbl 1047.20020
The authors call an automorphism of a finite group \(G\) a Coleman automorphism if its restriction to any Sylow subgroup of \(G\) is inner in \(G\). The authors show that the group of Coleman automorphisms modulo inner automorphisms \(\text{Out}_{\text{Col}}(G)\) is Abelian. Moreover, they show that if there is a prime \(p\) so that no composition factor of \(G\) has order \(p\), then \(p\) does not divide the order of \(\text{Out}_{\text{Col}}(G)\). In case no chief factor of \(G/O_2(G)\) has order \(2\), then the authors show that the normalizer of \(G\) in the units of the integral group ring \(\mathbb{Z} G\) equals the central units of the group ring multiplied with the group \(G\) itself. The consideration has its motivation in the spectacular counterexample to the isomorphism problem for integral group rings given by the first named author.

MSC:
20D45 Automorphisms of abstract finite groups
20E36 Automorphisms of infinite groups
16U70 Center, normalizer (invariant elements) (associative rings and algebras)
20C10 Integral representations of finite groups
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16S34 Group rings
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