Let $G$ be a finite group and $\sigma =\{ \sigma_{ i} | i\in I\}$ some partition of the set of all primes $\Bbb{ P}$, that is, $\sigma =\{ \sigma_{ i} | i\in I \}$, where $\Bbb{ P}=\bigcup_{ i\in I} \sigma_{ i}$ and $\sigma_{ i}\cap \sigma_{ j}= \emptyset $ for all $i\ne j$. We say that $G$ is $\sigma$-primary if $G$ is a $\sigma _{ i}$-group for some $i$. A subgroup $A$ of $G$ is said to be: ${ \sigma}$-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 $\sigma$-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 $\sigma$-quasinormal in $G$ if $L$ is modular and ${ \sigma}$-subnormal in $G$. We study $\sigma$-quasinormal subgroups of $G$. In particular, we prove that if a subgroup $H$ of $G$ is $\sigma$-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 $\sigma$-primary.
Categories: math.GR