On Supersolubility of a Group with Seminormal Subgroups

Сибирский математический журнал

article

V. S. Monakhov¹  A. A. Trofimuk² 

[1] Francisk Skorina Gomel State University;Pushkin Brest State University

关键词: supersoluble group;    nilpotent group;    seminormal subgroup;    derived subgroup;    X-residual;    index of a subgroup;    Sylow subgroup;   

DOI  :  10.1134/S0037446620010103

学科分类：数学（综合）

来源: Izdatel stvo Instituta Matematiki Rossiiskoi Akademii Nauk

【 摘 要 】

A subgroup A is called seminormal in a group G if there exists a subgroup B such that G = AB and AX is a subgroup of G for every subgroup X of B. Studying a group of the form G = AB with seminormal supersoluble subgroups A and B, we prove that $$G^{\mathfrak{U}}=\left(G^{\prime}\right)^{\mathfrak{N}}$$. Moreover, if the indices of the subgroups A and B of G are coprime then $$G^{\mathfrak{U}}=\left(G^{\mathfrak{N}}\right)^{2}$$. Here $$\mathfrak{N}$$, $$\mathfrak{U}$$, and $$\mathfrak{N}^2$$ are the formations of all nilpotent, supersoluble, and metanilpotent groups respectively, while $$H^{\mathfrak{X}}$$ is the $$\mathfrak{X}$$-residual of H. We also prove the supersolubility of G = AB when all Sylow subgroups of A and B are seminormal in G.

【 授权许可】

CC BY