Finite groups in which normality or quasinormality is transitive.
A `T-group' is a group in which normality is a transitive relation and a `PT-group' is one in which permutability is transitive. Recently there has been considerable interest in the problem of characterizing finite T-groups, especially in the soluble case. The characterizations usually take the form of restrictions on the Sylow structure of the group. In the work under review the author gives several new criteria for a finite group to be a soluble T-group or PT-group. The novelty here is that the restrictions only apply to Sylow \(p\)-subgroups of a group \(G\) for which the prime \(p\) divides the order of the generalized Fitting subgroup \(F^*(G)\). We quote two of the results obtained. (i)~A finite group \(G\) is a soluble T-group if and only if each \(p\)-subgroup is pronormal in \(G\) for all \(p\) dividing \(|F^*(G)|\). (ii)~A finite group \(G\) is a soluble PT-group if and only if \(H\leq P\in\text{Syl}_p(G)\) implies that \(H\) is permutable in \(N_G(P)\) for all \(p\) dividing \(|F^*(G)|\). (A subgroup \(H\) is `pronormal' in \(G\) if \(H\) and \(H^g\) are conjugate in \(\langle H,H^g\rangle\) for all \(g\in G\).)
scientific article; zbMATH DE number 2130536