Sylow 2-子群可补的有限群

On Finite Group with Sylow 2-Subgroup Complemented

设G是有限群,H≤G.如果G中存在子群K≤G满足G=KH,且H∩K=1,那么称H在G中可补.通过研究G的Sylow2-子群的可补性,证明了:设G为有限群,|G|=2^at,(2,t)=1,若G的Sylow2-子群可补且G是PSL2(p^r)-自由的,p^r=2^a-1,其中p为素数,r为正整数,则G可解.

Let G be a finite group,and H be a subgroup of G.If there exists a subgroup K of G such that G=KH and H∩K=1,then H is called complemented in G.We get the following results: Let G be a finite group,|G|=2^at,(2,t)=1,if Sylow 2-subgroup of G is complemented and G is PSL 2(p^r)-free,p r=2^a-1,where p is prime,r is positive integer,then G is solvable.