The Influence Of General Permutability On The Structure Of Finite Groups

Many scholars have studied the influence of general normality on several kinds of structures of finite groups,especially on p-nilpotency,nilpotency and supersolvability.In this dissertation,two aspects are researched,one is the effect of general normality of minimal subgroups,the other is the effect of general normality of maximal subgroups of some subgroups.Firstly,we investigate the influence of general normality of minimal subgroups on the structure of finite group.In the second chapter,we research the relationship between the p-nilpotency of G and s-semipermutability or N E-property of the cyclic subgroups with order p or 4,where p is the smallest prime dividing the order of finite group G.That's to say,it is followed that G is p-nilpotent if cyclic subgroups of G with order p and 4 are either s-semipermutable subgroups or N E-subgroups.Based on this,we generalize above result to the situation of G=AB,where A and B are subnormal subgroups of G,or A and B are totally permutable.In the third chapter,we apply the concept of weakly SS-semipermutable subgroups to study the influence of cyclic subgroups of order p or 4 of(general)Fitting subgroups of finite group G on the supersolvability and nilpotency,then we obtain some sufficient or necessary conditions on supersolvability(nilpotency)of finite groups.Moreover,we generalize above results to saturated formations.Secondly,we research the influence of general normality of maximal subgroups of some subgropups on the structure of finite groups.In the fourth chapter,we research the effect of CSS-permutability of maximal subgroups of Sylow subgroups of(general)Fitting subgroup on the supersolvability of finite groups.And then we obtain some sufficient conditions on the supersolvability of finite groups.Furthermore,we extend our results to saturated formations.