On permutizers of subgroups of finite groups

Abstract

Let H be a subgroup of a group G. The permutizer of H in G is the subgroup PG(H) = hx ∈ G | hxiH = Hhxii. A subgroup H of a group G is called permuteral in G, if PG(H) = G, and strongly permuteral in G, if PU(H) = U whenever H ≤ U ≤ G. Finite groups with given systems of permuteral and strongly permuteral subgroups are studied. New characterizations of w-supersoluble and supersoluble groups are received.