Structure of finite groups with some weakly $S$-semipermutable subgroups

Document Type : Research Paper

Authors

1 Department of Mathematics, payme Noor University, P. O. Box 19395-4697, Tehran, IRAN

2 Faculty of Mathematical Sciences, University of Shahrekord, Shahrekord, Iran.

3 Department of Mathematics, University of Payme Noor, P.O.Box 19395-4697, Tehran, Iran

Abstract

Let $ G $ be a finite group. If $ A\leq G $, recall that $ A $ is  weakly $S$-semipermutable  in $G$ provided there is $K\unlhd G$ such that   $KA$ is $S$-permutable in $G$, and  $K\cap A$ is $S$-semipermutable in $G$. The purpose of this paper is to demonstrate that weakly $S$-semipermutability of special types of subgroups in a finite group $ G $ can help us to determine  structural properties of $ G $. For example, given a prime $p$, a $p$-soluble finite group $G$ and a Sylow $p$-subgroup $G_{p}$ of $G$, we will show that $G$ is $p$-supersoluble if the maximal subgroups of $G_{p}$ are weakly $S$-semipermutable in $ G$. Moreover, we use the concept of weakly $S$-semipermutability to prove new criteria for $p$-nilpotency of finite groups.

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References

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Journal of Mahani Mathematical Research
Volume 13, Issue 1 - Serial Number 26
November 2023
Pages 457-466

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History

  • Receive Date: 19 February 2023
  • Revise Date: 27 June 2023
  • Accept Date: 17 October 2023

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