n-极小子群、n-极大子群与有限群结构研究

This project mainly studies the relationship between n-minimal or n-maximal subgroups and the structure of a finite group. Details are as follows:.(1) Assume that all n-minimal or n-maximal subgroups of a finite group are S-quasinormal. We will find the boundary value of the maximal length of subgroup-chains for the solvability. Furthermore, we will explore the nilpotent length..(2) We will describe the structure when the group has unique n-minimal subgroup..(3) To prove or find a counterexample of the conjecture that a finite group G is supersolvable if and only if the number of prime factors for every maximal subgroup index is equal.

本项目主要研究n-极小子群(或n-极大子群)与有限群结构之间的联系。具体如下：.(1)、在假定有限群的所有n-极小子群(或n-极大子群)S-拟正规的条件下，找到用极大链长来刻画群可解性时的临界值。进一步给出幂零长信息。.(2)、在假定有限群的n-极小子群唯一的条件下，给出群的结构。.(3)、证明猜想或给出反例：群G超可解当且仅当G的所有极大子群指数的素因子个数一样多。

设k(G)为有限群G的共轭类个数，Irr(G)为G的复不可约特征标集合，T(G)为群G的所有复不可约特征标次数和，Irr_p(G)={χ∈Irr(G)|χ(1)=1或p|χ(1)}，T_p(G)为Irr_p(G)中的复不可约特征标次数和，Irr_p'(G)={χ∈Irr(G)| p∤χ(1)}，T_(p^')(G)为Irr_p'(G)中的复不可约特征标次数和，这里p为素数。本项目得到了用与共轭类个数k(G)，特征标次数和T(G)以及部分和T_p(G)，T_(p^')(G)的相关的量来刻画有限群结构（p-超可解性，p-幂零性，p-闭性）的深刻结果。