# Knyagina V. N.

### On the solvability of a finite group with $S$-seminormal Schmidt subgroups

Knyagina V. N., Monakhov V. S., Zubei E. V.

↓ Abstract

Ukr. Mat. Zh. - 2018. - 70, № 11. - pp. 1511-1518

A finite nonnilpotent group is called a Schmidt group if all its proper subgroups are nilpotent. A subgroup $A$ is called $S$-seminormal (or $SS$-permutable) in a finite group $G$ if there is a subgroup B such that $G = AB$ and $A$ is permutable with every Sylow subgroup of B. We establish the criteria of solvability and $\pi$ -solvability of finite groups in which some Schmidt subgroups are $S$-seminormal. In particular, we prove the solvability of a finite group in which all supersoluble Schmidt subgroups of even order are $S$-seminormal.

### On the Derived Length of a Finite Group with Complemented Subgroups of Order $p^2$

Knyagina V. N., Monakhov V. S.

Ukr. Mat. Zh. - 2015. - 67, № 7. - pp. 874–881

It is shown that a finite group with complemented subgroups of order $p^2$ is soluble for all $p$ and its derived length does not exceed 4.

### Factorizations of Finite Groups into $r$-Soluble Subgroups with Given Embeddings

Knyagina V. N., Tyutyanov V. N.

Ukr. Mat. Zh. - 2014. - 66, № 10. - pp. 1431–1435

Let $X$ be a subset of the set of positive integers. A subgroup $H$ of a group $G$ is called $X$-subnormal in $G$ if there exists a chain of subgroups $H = H_0 ⊆ H_1 ⊆ … ⊆ H_n = G$ such that $|H_i : H_{i-1}| ∈ X$ for all $i$. We study the solubility and $r$ -solubility of a finite group $G = AB$ with some restrictions imposed on the subgroups $A$ and $B$ and on the set $X$ .