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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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