π－拟正规子群

π-QUASI-NORMAL SUBGROUPS

本文讨论有限群的π－拟正规子群的性质及其对群结构的影响。

This paper discusses the properties of the π-quasi-normal subgroups of a finite group G, and investigates how the structure of G is influenced by the π-quasi-normal subgroups of G. The major conclusions are following: (1) Gis a π-quasi-nilpotent group, if every maximal subgroup of Gis π-quasi-normal in G. (2) every 2-maximal subgroup of C is π-quasi-normal in G, if G is a π-quasi-nilpotent group, or a q -basic group of order paq.(3) If all maximal subgroups of every Sylow subgoup of G are π-quasi-normal in G, then G is a supersolvable group whose every Sylow subgroup is a s -semi-normal subgroup of G. (4)p∈,π(G),if there exists a p'-Hall subgroup of G whose all maximal subgroups are p'-quasi-normal in G, then G belongs to one class of the following three classes of groups: (i) nilpotent groups; (ii) the groups which are extensions of p -groups by cyclic q -groups; (iii) the groups, with order pαqβ, whose every Sylow subgroup is cyclic.