Maximal subgroups and PST-groups

A subgroup H of a group G is said to permute with a subgroup K of G if HK is a subgroup of G. H is said to be permutable (resp. S-permutable) if it permutes with all the subgroups (resp. Sylow subgroups) of G. Finite groups in which permutability (resp. S-permutability) is a transitive relation are called PT-groups (resp. PST-groups). PT-, PST- and T-groups, or groups in which normality is transitive, have been extensively studied and characterised. Kaplan [Kaplan G., On T-groups, supersolvable groups, and maximal subgroups, Arch. Math. (Basel), 2011, 96(1), 19-25] presented some new characterisations of soluble T-groups. The main goal of this paper is to establish PT- and PST-versiosn of Kaplan's results, which enables a better understanding of the relationships between these classes

On a class of generalised Schmidt groups

In this paper families of non-nilpotent subgroups covering the non-nilpotent part of a finite group are considered. An $A_5$-free group possessing one of these families is soluble, and soluble groups with this property have Fitting length at most three. A bound on the number of primes dividing the order of the group is also obtained