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Class preserving automorphisms of Blackburn groups. (English) Zbl 1102.20023
Let $$G$$ be a Blackburn group, i.e., a finite group such that $$\bigcap_UU\neq 1$$ where $$U$$ runs through the nonnormal subgroups of $$G$$. Then the normalizer of $$G$$ in the unit group $$(\mathbb{Z} G)^\times$$ of its integral group ring $$\mathbb{Z} G$$ is the product of $$G$$ with the central units of $$(\mathbb{Z} G)^\times$$ [Y. Li, S. K. Sehgal and M. M. Parmenter, Commun. Algebra 27, No. 9, 4217-4233 (1999; Zbl 0943.16012)].
The paper under review improves on this by showing that the class preserving automorphisms of $$G$$ are inner automorphisms. As a corollary, if $$H$$ is a torsion group which has a nonnormal subgroup, and if $$R$$ is an integral domain of characteristic zero in which no order of the elements in $$H$$ is invertible, then $$u\in Z((RH)^\times)\cdot H$$ holds for every unit $$u\in(RH)^\times$$ which commutes with all unipotent elements of $$\mathbb{Z} H$$. – The proofs often refer to M. Hertweck’s habilitation thesis in which these results also appear except for a special type of Blackburn groups.

##### MSC:
 20D45 Automorphisms of abstract finite groups 16U60 Units, groups of units (associative rings and algebras) 16S34 Group rings 20C05 Group rings of finite groups and their modules (group-theoretic aspects)
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##### References:
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