Finite Groups with $H_σ$-Permutably Embedded Subgroups

Abstract

Let $G$ be a finite group. Let $\sigma=\{\sigma_i|i\inI\}$ be a partition of the set of all primes $P$ and $n$ an integer. We write $\sigma(n) = \{\sigma_i|\sigma_i\cap \pi(n)\neq\emptyset\}$, $\sigma(G) = \sigma(|G|)$. A set $H$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every member of $H\backslash \{1\}$ is a Hall $\sigma_i$-subgroup of $G$ for some $\sigma_i$ and $H$ contains exact one Hall $σ_i$-subgroup of $G$ for every $σ_i \in \sigma(G)$. A subgroup $A$ of $G$ is called: (i) a $\sigma$-Hall subgroup of $G$ if $\sigma(|A|) \cap \sigma(|G : A|) = \emptyset$; (ii) $\sigma$-permutable in $G$ if $G$ possesses a complete Hall $\sigma$-set $H$ such that $AHx =HxA$ for all $H\in H$ and all $x\in G$. We say that a subgroup $A$ of $G$ is $H_σ$-permutably embedded in $G$ if $A$ is a $\sigma$-Hall subgroup of some $\sigma$-permutable subgroup of $G$. We describe the structure of $G$ assuming that every subgroup of $G$ is $H_σ$-permuta- bly embedded in $G$.

Keywords

• σ-Hall subgroup
• σ-subnormal subgroup
• σ-nilpotent group