Schmidt子群为Hall S-拟正规嵌入群的有限群

On Finite Groups with Hall S-Quasinormally Embedded Schmidt Subgroups

设H是有限群G的子群.如果H为G的S-拟正规闭包H^(sqG)的Hall子群,则称H为G的一个Hall S-拟正规嵌入子群.如果一个非幂零有限群的任一真子群幂零,则称这个非幂零群为Schmidt群.该文证明了:如果有限群G的每一个Schmidt子群均为G中Hall S-拟正规嵌入子群,则G′幂零.

A subgroup H of a finite group G is said to be Hall S-qusinormally embedded in G,if H is also a Hall subgroup of some S-quasinormal subgroup of G.A Schmidt group is a non-nilpotent finite group whose all proper subgroups are nilpotent.In this paper,it has been proved that if each Schmidt subgroup of a finite group G is Hall S-qusinormally embedded in G,then the derived subgroup G′is nilpotent.