The structure of finite groups in which permutability is a transitive relation (Q2726684)

A subgroup \(H\) of a group \(G\) is called permutable if \(HX=XH\) for all subgroups \(X\) of \(G\), and a group \(G\) is said to be a PT-group if permutability in \(G\) is a transitive relation. Thus PT-groups can be considered as the natural generalization corresponding to T-groups when normality is replaced by permutability. The structure of soluble PT-groups has been investigated by \textit{J. C. Beidleman, B. Brewster} and \textit{D. J. S. Robinson} [J. Algebra 222, No. 2, 400-412 (1999; Zbl 0948.20015)]. In the paper under review the author deals with finite insoluble PT-groups. It is proved that every finite PT-group is an SC-group, i.e. all its chief factors are simple. A structure theorem for finite PT-groups is obtained, and the behaviour of Abelian chief factors of finite PT-groups is studied. Moreover, groups generated by subnormal PT-subgroups are described.