子群的π-可补性对群结构的影响

The influence of π-supplemented subgroups on the structure of finite groups

如果存在G的一个子群K,使得G=HK且|H∩K|π=1,则群G的一个子群H称为在G中π-可补,此时K称为H在G中的π-补.研究了π-可补子群的一些性质,并利用群G的Sylowp-子群的极大和极小子群的π-可补性,给出了群G为p-幂零群的一些条件.特别地证明了如下结果:设G是一个群,P是G的一个Sylowp-子群,p∈π且p是|G|的一个素因子,如果(|G|,p-1)=1且P的每个极大子群在G中π-可补,则G是p-幂零群.

If there exists a subgroup K of G such that G=HK and |H∩K|_π=1, then a subgroup H of G is said to be π-supplemented in G. The authors give some sufficient conditions under which a group G is p-nilpotent by using π-supplementility of maximal subgroup and minimal subgroup of Sylow p-subgroup of G. Particularly, prove the following result: Let G be a group, p∈π and p be a prime divisor of |G|. If (|G|,p-1)=1 and every maximal subgroups of Sylow p-supgroup of G is π-supplemented in G, then G is p-nilpotent.