The Influence of Weakly ${\sigma }$-Permutably Embedded Subgroups on the Structure of Finite Groups

Abstract

Let $\sigma =\{{\sigma _i|i\in I}\}$ be some partition of the set of all primes ${\mathbb {P}}$, G a finite group and $\sigma (G)=\{{\sigma _i|\sigma _i \cap \pi (G) \ne \emptyset }\}$. A set ${\mathcal {H}}$ of subgroups of G is said to be a complete Hall $\sigma$-set of G if every non-identity member of ${\mathcal {H}}$ is a Hall $\sigma _i$-subgroup of G and ${\mathcal {H}}$ contains exactly one Hall $\sigma _i$-subgroup of G for every $\sigma _i\in \sigma (G)$. G is said to be $\sigma$-full if G possesses a complete Hall $\sigma$-set. A subgroup H of G is said to be $\sigma$-permutable in G provided there is a complete Hall $\sigma$-set ${\mathcal {H}}$ of G such that $HA^x=A^xH$ for all $A\in {\mathcal {H}}$ and all $x\in G$; $\sigma$-permutably embedded in G if H is $\sigma$-full and for every $\sigma _i \in \sigma (H)$, every Hall $\sigma _i$-subgroup of H is also a Hall $\sigma _i$-subgroup of some $\sigma$-permutable subgroup of G. We call that a subgroup H of G is weakly ${\sigma }$-permutably embedded in G if there exists a $\sigma$-subnormal subgroup T of G such that $G=HT$ and $H\cap T\le H_{\sigma eG}$, where $H_{\sigma eG}$ is the subgroup of H generated by all those subgroups of H which are $\sigma$-permutably embedded in G. In this paper, we study the structure of G under the condition that some given subgroups of G are weakly ${\sigma }$-permutably embedded in G. Some known results are generalized.