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Class-preserving Coleman automorphisms of finite groups. (English) Zbl 1004.20011
Let $$G$$ be a finite group with dihedral or generalized quaternion Sylow 2-subgroup. Let $$A$$ be the group of automorphisms of $$G$$ which preserve conjugacy classes of $$G$$ and whose restriction to any Sylow subgroup is given by conjugation with an element in $$G$$. The author shows that then the quotient of $$A$$ by the group of inner automorphisms of $$G$$ is of odd order. The interest in this type of questions mainly comes from the author’s recent spectacular counterexample to the isomorphism problem for integral group rings. There, this class of automorphisms is intensively used. The proof of the present paper also is a followup of the author’s sophisticated technique which he developed for his example.

##### MSC:
 20D45 Automorphisms of abstract finite groups 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure 16S34 Group rings 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 20E45 Conjugacy classes for groups
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