On some classes of supersoluble groups

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