Some Sufficient Conditions for Supersolubility of Groups

The main purpose of the present paper is to prove the following some theorems: for a finite group G,if G satisfies one of the following conditions,then G is supersolvable;(1) A maximal and cyclic subgroup of G is weakly quasi-normal in G.(2) All Sylow subgroups and all maximal subgroups of Sylow subgroups of a subgroup which has index power of a prime of G are weakly quasi-normal in G.(3) All maximal subgroups of a maximal subgroup of G which is soluble are weakly quasi-normal in G.(4) All cyclic subgroups of Sylow subgroups of G which is soluble are weakly quasi-normal in G.(5) All maximal subgroups of Sylow subgroups of G which is soluble are weakly quasi-normal in G.(6) Let G be the products of A which is a Hall-π subgroup of G and B a Hall-π′ subgroup.All Sylow subgroups of A and B are weakly quasi-normal in G.