The Influence Of S-quasi-normal, Conditional Permution To The Supersolvable Groups

Supersolvable subgroup plays an important role in the study of fnite group.People are trying to investigate the supersolvable groups from the structure of thegroups and the certain subgroups. We characterize the structure of fnite groupsand their properties, and get lots of useful results.Departuring from the basic defnition and lemma, I emphatically fnish somesupersolvable groups theorems from the group theory scholars, including conditionalpermution、completely conditional permution、conjugate-permutable subgroups、s-quasi-normal subgroups on the group supersolvablility, and then writing down ownresearch results. Investigating the conditional permution of supersolvable groupsfrom the maximal cyclic-subgroup of G. I am from the Sylow subgroups, maximalsubgroups and between subgroups of completely conditional permution to describethe groups, through the weakly condition of supersolvable group obtain some suf-cient conditions. Let G is a fnite group, G is non-cyclic subgroup of maximal cyclicsubgroup and CG(A)=NG(A) or A￠G and every cyclic subgroup of G is conditionreplacement in G, then G is the supersolvable groups; Let G=HK,(|H|,|K|)=1,H is the nilpotent groups, K is the supersolvable groups, H￠G, the Sylow sub-groups and their maximal subgroups of H is the condition replacement with K,then G is the supersolvable groups. The main method is used by a minimal coun-terexample method. Summarizing the main content of this paper and obtainingsome sufcient conditions for supersolvable groups.