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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.

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)
Full Text: DOI
[1] DOI: 10.1016/0022-4049(87)90028-4 · Zbl 0624.20024 · doi:10.1016/0022-4049(87)90028-4
[2] Huppert, Endliche Gruppen I (1967) · Zbl 0217.07201 · doi:10.1007/978-3-642-64981-3
[3] DOI: 10.1080/00927879908826692 · Zbl 0943.16012 · doi:10.1080/00927879908826692
[4] DOI: 10.1016/0021-8693(66)90018-4 · Zbl 0141.02401 · doi:10.1016/0021-8693(66)90018-4
[5] Hertweck, Contributions to the integral representation theory of groups (2004) · Zbl 1069.20005
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