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Some characterizations of finite groups in which semipermutability is a transitive relation. (English) Zbl 1205.20025

Forum Math. 22, No. 5, 855-862 (2010); corrigendum ibid. 24, No. 6, 1333-1334 (2012).
Let \(G\) be a finite group and let \(H\) a subgroup of \(G\); \(H\) is said to be semipermutable in \(G\) if \(H\) permutes with every subgroup of \(G\) with \((|H|,|K|)=1\). A group \(G\) is called a BT-group if semipermutability is a transitive relation in \(G\).
The main aim of this paper is to provide some new characterizations of finite solvable BT-groups. In particular, it is proved (Theorem C) that a finite group \(G\) is a solvable BT-group if and only if it has a normal subgroup \(N\) such that \(N\) and \(G/N''\) are solvable BT-groups.

MSC:

20D40 Products of subgroups of abstract finite groups
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D35 Subnormal subgroups of abstract finite groups
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