有限群的σ半次正规子群

Onσ-Semisubnormal Subgroups of Finite Groups

令G是一个有限群.如果G中存在子群K,满足G=HK,且对任一K_(1)<K有HK_(1)<G,则称H在G中是半正规的.本文结合子群的σ次正规性和半正规性引入了新的子群概念.如果群G的子群H在G中是σ次正规的或者半正规的,则称H在G中是σ半次正规的.通过研究群G的完全Hallσ集中的子群及其极大子群的σ半次正规性,给出了G是σ可解群和超可解群的若干新的判别准则.

G is a finite Group.A subgroup H of G is said to be seminormal in G if there is a subgroup K in G such that G=HK and the product HK_(1) is a proper subgroup of G for any subgroup K_(1) in K which differs from K.In this paper,a new concept of subgroups is introduced by combining theσ-subnormality and seminormality of subgroups.A subgroup H of G is calledσ-semisubnormal in G,if H isσ-subnormal in G or seminormal in G.By studying theσ-seminormality of subgroups and their maximal subgroups in the complete Hall set of typeσof G,some new criteria for G to be solvable group and supersolvable group are given.