The Influence Of CC-subgroups, C-supplementarity Of Minimal Subgroups On The Structure Of Finite Groups

Let H be a subgroup of a finite group G. H is called a CC-subgroup of G, if for any 1≠x∈H, C_G(x)≤H.Let G be a finite group. A subgroup H of G is said to be c-supplemented in G if there exists a subgroup K of G such that G = HK, H∩K≤H_G = Core_G(H).Firstly, this paper introduces some properties on cc-subgroupsand and discusses the finite groups which contain proper cc-subgroup. Under the famous theorem of Schur-Zassenhaus ,we prove the following theorem: Let H be a solvable and normal CC-subgroup of G. Then (1)H is nilpotent and has a complement in G and all such complements are CC-subgroups of G. (2)G is a Frobenius group.Secondly, in this paper,we study the solvability and p-nilpotency of finite groups under the assumption that some minimal subgroups are c-supplemented. In [15],Guo and Shum proved the following theorem : Let G be a finite group and Let P be a Sylow p-subgroup of G where p is a minimal prime disvisor of |G|. Suppose that every minimal subgroup of P∩G~N is c-supplemented in G, and when p = 2, either every cyclic subgroup of order 4 of P∩G~N is c-supplemented in G or P is quaternion-free, where G~N is the N -residue of G. In this paper we prove that G is solvable under the assumption that the subgroup is c-supplemented in G'. We also prove G is p-nilpotent under the assumption that the subgroup is c-supplemented in O~p(G).