On some classes of supersoluble groups

Finite groups G for which for every subgroup H and for all primes q dividing the index I G : H I there exists a subgroup K of G such that H is contained in K and vertical bar K : H vertical bar = q are called Y-groups. Groups in which subnormal subgroups permute with all Sylow subgroups are called PST-groups. In this paper a local version of the Y-property leading to a local characterisation of Y-groups, from which the classical characterisation emerges, is introduced. The relationship between PST-groups and Y-groups is also analysed. (C) 2006 Elsevier Inc. All rights reserved.