On supersolvable groups and the nilpotator

A finite group G is called G a J-group if each subnormal subgroup of G is normal in G and a subgroup K of G is called an ℋ -subgroup of G if N _G (K) n K^g ∩ K^g⊆ K for all g εG. Using the notion of ℋ-subgroups, we present some new conditions for supersolvability and we characterize supersolvable groups, which are either J-groups or A-groups (i.e., all their Sylow subgroups are abelian). For example, we prove that if all cyclic subgroups of G of prime order or of order 4 are ℋsubgroups of G, then G is supersolvable with a well defined structure. We also show, that an A-group G is supersolvable if and only if its Sylow subgroups are products of cyclic ℋ-subgroups of G.