The Influence Of Normal Index And S-normality Of Subgroups On The Structure Of Finite Groups

Let H be a proper subgroup of a finite group G, M is a normal subgroup of G,we call |M/M∩H_G| to be the normal index of H in G, if M satisfying (1)G=MH;(2) if K(?)G and G=KH, then |K/K∩H_G|≥|M/M∩H_G|. We writeη^*(G:H) forthe normal index of H in G.A subgroup H of a finite group G is called s-normal in G, if there exists a subnor-mal subgroup K of G such that G=HK and H∩K≤H_SG, where H_SG is the largestsubnormal subgroup of G which is contained in H.The main results of this paper as follow:(1) We obtain some results on solvability, p-solvability and p-nilpotency of finitegroups by the number theory property of normal index of 2-maximal subgroups andSylow subgroups.(2) We obtain some sufficient conditions on solvability of finite groups by thes-normality of Sylow normalizers.