# Miao L.

### Some conditions for cyclic chief factors of finite groups

↓ Abstract

Ukr. Mat. Zh. - 2016. - 68, № 12. - pp. 1718-1722

A subgroup $H$ of a finite group $G$ is called $\scrM$ -supplemented in $G$ if there exists a subgroup $B$ of $G$ such that $G = HB$ and $H_1B$ is a proper subgroup of $G$ for every maximal subgroup $H_1$ of $H$. The main purpose of the paper is to study the influence of $\scrM$ -supplemented subgroups on the cyclic chief factors of finite groups.

### On Nearly *ℳ*-Supplemented Subgroups of Finite Groups

Ukr. Mat. Zh. - 2014. - 66, № 1. - pp. 63–70

A subgroup *H* is called nearly *ℳ*-supplemented in a finite group *G* if there exists a normal subgroup *K* of *G* such that *HK* ⊴ *G* and *TK* < *HK* for every maximal subgroup *T* of *H.* We obtain some new results on supersoluble groups and their formation by using nearly *ℳ*-supplemented subgroups and study the structure of finite groups.

*Q* -permutable subgroups of finite groups

Ukr. Mat. Zh. - 2011. - 63, № 11. - pp. 1534-1543

A subgroup $H$ of a group $G$ is called $Q$-permutable in $G$ if there exists a subgroup $B$ of $G$ such that (1) $G = HB$ and (2) if $H_1$ is a maximal subgroup of $H$ containing $H_{QG}$, then $H_1B = BH_1 < G$, where $H_{QG}$ is the largest permutable subgroup of $G$ contained in $H$. In this paper we prove that: Let $F$ be a saturated formation containing $U$ and $G$ be a group with a normal subgroup $H$ such that $G/H \in F$. If every maximal subgroup of every noncyclic Sylow subgroup of $F∗(H)$ having no supersolvable supplement in $G$ is $Q$-permutable in $G$, then $G \in F$.