The Influence Of Conditionally Permutable Subgroups And S-Normal Subgroups On The Structure Of Finite Groups

Groups mainly study the mathematical theory of the structure of fnite groupsand the properties of their subgroups. The structure of fnite groups is closelyrelated to the proterties of some special subgroups. If we study the properties ofdiferent subgroups, we can get diferent structure of groups. A subgroup H of afnite group G is called quasi-normal or permutable if for any T satisfying T≤G,there holds HT=T H. A subgroup H of a fnite group G is called conditionallypermutable subgroup if for any subgroup K of G, there is an element x of G such thatHK^x=K^xH. A subgroup H of a fnite group G is called completely conditionallypermutable subgroup if for any subgroup K of G, there is an element y of H, K such that HK^y=K^yH. We called a subgroup H of a fnite group G s-normal inG, if there is a subnormal subgroup K of G such that G=HK and H∩K≤H_SG,where H_SGis the maximal subnormal subgroup of G which is contained in H. Ormay be defned as: a subgroup H of a fnite group G is called s-normal in G, ifthere is a subnormal subgroup K of G such that G=HK and H∩K G.This article mainly uses conditionally permutable subgroups, completely con-ditionally permutable subgroups and s-normal subgroups to study the infuence onfnite nilpotent groups, fnite soluble groups and fnite p-nilpotent groups, gettingthe following main new conclusions:（1） Let G be a fnite group, if the group Gsatisfes the following conditions:（ⅰ） There is a index for p nilpotent maximal sub-group M of G, in which p=max（π（G））;（ⅱ） There is a Sylow p-subgroup P of Gsuch that NG（P）/CG（P） is a p-group, and P and the cycle T subgroup of G areconditionally permutable, then G is a nilpotent group.（2） Let G be a fnite group,if CI-sections of every maximal subgroup of G are s-normal, then G is a solvablegroup.（3） Let G be a fnite group, p is the least prime factor of G, P∈Sylp（G）. Ifevery maximal subgroup P s-normal to G, then G is a p-nilpotent group.