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Automorphisms of finite groups. (English) Zbl 1072.20030
For a finite group \(G\), the authors define \(\text{Out}_c(G)=\operatorname{Aut}_c(G)/\text{Inn}(G)\), where \(\operatorname{Aut}_c(G)\) stands for the group of the so-called Coleman automorphisms of \(G\). A Coleman automorphism (C-automorphism for short) is \(\alpha\in\operatorname{Aut}(G)\) such that \(\alpha\) is a class-preserving automorphism, \(\alpha\) becomes inner on Sylow \(p\)-subgroups and \(\alpha^2\in\text{Inn}(G)\).
The paper contains a number of sufficient conditions for \(\text{Out}_c(G)\) to be trivial. This is shown to happen provided that: 1) \(G\) is nilpotent-by-nilpotent with generalized quaternion Sylow 2-subgroups. 2) \(G\) is metabelian with dihedral Sylow 2-subgroups. 3) \(G\) has Abelian Sylow 2-subgroups and there exists a proper nilpotent normal subgroup \(N\) of \(G\) such that \(G/N\) has a normal Sylow 2-subgroup.

MSC:
20D45 Automorphisms of abstract finite groups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20C05 Group rings of finite groups and their modules (group-theoretic aspects)
16U60 Units, groups of units (associative rings and algebras)
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