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.