Let H be a subgroup of a group G. The permutizer \(P_G(H)\) is the subgroup generated by all cyclic subgroups of G which permute with H. A subgroup H of a group G is strongly permuteral in G if \(P_U(H)=U\) for every subgroup U of G, such that \(H\le U\le G\) . We investigate groups with \(\mathbb {P}\) -subnormal or strongly permuteral Sylow subgroups. Moreover, we prove that groups with all strongly permuteral primary cyclic subgroups are supersoluble.
