On $σ$-quasinormal subgroups of finite groups

Abstract

Let $G$ be a finite group and $σ=\{σ_{i} | i\in I\}$ some partition of the set of all primes $\Bbb{P}$, that is, $σ=\{σ_{i} | i\in I \}$, where $\Bbb{P}=\bigcup_{i\in I} σ_{i}$ and $σ_{i}\cap σ_{j}= \emptyset $ for all $i\ne j$. We say that $G$ is $σ$-primary if $G$ is a $σ_{i}$-group for some $i$. A subgroup $A$ of $G$ is said to be: $σ$-subnormal in $G$ if there is a subgroup chain $A=A_{0} \leq A_{1} \leq \cdots \leq A_{n}=G$ such that either $A_{i-1}\trianglelefteq A_{i}$ or $A_{i}/(A_{i-1})_{A_{i}}$ is $σ$-primary for all $i=1, \ldots, n$, modular in $G$ if the following conditions hold: (i) $\langle X, A \cap Z \rangle=\langle X, A \rangle \cap Z$ for all $X \leq G, Z \leq G$ such that $X \leq Z$, and (ii) $\langle A, Y \cap Z \rangle=\langle A, Y \rangle \cap Z$ for all $Y \leq G, Z \leq G$ such that $A \leq Z$. In this paper, a subgroup $A$ of $G$ is called $σ$-quasinormal in $G$ if $L$ is modular and $σ$-subnormal in $G$. We study $σ$-quasinormal subgroups of $G$. In particular, we prove that if a subgroup $H$ of $G$ is $σ$-quasinormal in $G$, then for every chief factor $H/K$ of $G$ between $H^{G}$ and $H_{G}$ the semidirect product $(H/K)\rtimes (G/C_{G}(H/K))$ is $σ$-primary.

9 pages