非正规极大子群的迹对群可解性的影响

Influence of Traces of Non-Normal Maximal Subgroups on Solvability of Finite Groups

摘要: 在有限群论中，子群的性质是刻画群可解性的重要工具。本文利用非正规极大子群的迹的幂零性研究了可解群的结构，得到了一个关于可解群的充分必要条件(有限群G是可解的当且仅当G的每个非正规极大子群有幂零的迹)，推广了已知结果。

Abstract: In finite group theory, the properties of subgroups are an important tool to characterize the solv-ability of groups. In this paper, we study the structure of solvable groups by using the nilpotent property of traces of non-normal maximal subgroups, and obtain a necessary and sufficient condi-tion for solvable groups (A finite group G is solvable if and only if every non-normal maximal sub-group of G has a nilpotent trace), and generalize the known results.