# On *σ*-Embedded and *σ*-*n*-Embedded Subgroups of Finite Groups

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## Abstract

Let *G* be a finite group, and let *σ* = {*σ*_{i} | *i* ∈ *I*} be a partition of the set of all primes ℙ and *σ*(*G*) = {*σ*_{i} | *σ*_{i} ∩ *π*(*G*) ≠ ∅}. A set ℋ of subgroups of *G* is said to be a complete Hall *σ*-set of *G* if each nonidentity member of ℋ is a Hall *σ*_{i}-subgroup of *G* and ℋ has exactly one Hall *σ*_{i}-subgroup of *G* for every *σ*_{i} ∈ *σ* (*G*). A subgroup *H* of *G* is said to be *σ*-permutable in *G* if *G* possesses a complete Hall *σ*-set ℋ such that *HA*^{x} = *A*^{x}*H* for all *A* ∈ ℋ and *x* ∈ *G*. A subgroup *H* of *G* is said to be *σ*-*n*-embedded in *G* if there exists a normal subgroup *T* of *G* such that *HT* = *H*^{G} and *H* ∩ *T* ≤ *H*_{σG}, where *H*_{σG} is the subgroup of *H* generated by all those subgroups of *H* that are *σ*-permutable in *G*. A subgroup *H* of *G* is said to be *σ*-embedded in *G* if there exists a *σ*-permutable subgroup *T* of *G* such that *HT* = *H*^{σG} and *H* ∩ *T* ≤ *H*_{σG}, where *H*^{σG} is the intersection of all *σ*-permutable subgroups of *G* containing *H*. We study the structure of finite groups under the condition that some given subgroups of *G* are *σ*-embedded and *σ*-*n*-embedded. In particular, we give the conditions for a normal subgroup of *G* to be hypercyclically embedded.

## Keywords

finite group*σ*-embedded subgroup

*σ*-n-embedded subgroup supersoluble hypercyclically embedded

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