The Influence Of Subgroup's Complementarity On The Structure Of Finite Groups

In this paper we mainly study subgroup's complementarity of finite groups.Let G be a finite group.H?G.If there exists a subgroup K?G.such that G=HK.and H ? K=1.then H is called complemented in G.In group theory study,study-ing the influence of subgroup's complementarity on the structure and properties of finite groups is a very important topic.On this subject we obtain three results:First we characterize the simple group A5 by the number of uncomplemented subgroups;Then we study the structure and properties of finite groups whose subgroups of order 8 are complemented in detail;Finally we discuss the important influence of complementarity of Sylow 2-subgroups on the solvability of finite groups further.The main conclusions are as follows:Theorem 3.6 Let G be a group of order 60 whose Sylow 2-subgroup is elemen-tary abelian 2-subgroup of order 4.If there exists subgroup of order 2 of G that is uncomplemented in G,then G?A5=or G?<A4×C5.Theorem 3.7 Suppose |G|=60;G?A5 if and only if G only has 46 uncom-plemented subgroups.Theorem 4.6 Let G be a finite group whose subgroups of order 8 are comple-mented in G.If G is non-abelian simple group,then G?PSL2(7).Theorem 5.2 If every Sylow 2-subgroup of finite group G is abelian and is complemented in G,then G is solvable.Theorem 5.4 Let G be a non-abelian finite simple group,where |G|=2at,(2,t)=1,then every Sylow 2-subgroup of G is complemented in G if and only if G?PSL2(q),q=2a-1.Theorem 5.5 Let G be a finite group,|G|=2at,(2,t)=1.If Sylow 2-subgroup of G is complemented in G and G is PSL2(pr)-free,pr=2a-1,where p is prime,r is positive integer,then G is solvable.Theorem 5.6 Let G be a non-abelian finite simple group and 7 |G|.If Sylow 7-subgroup of G is complemented in G and G is PSL3(q)-free,then G?A7.