Journal of Mahani Mathematical Research

Journal of Mahani Mathematical Research

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

Document Type : Research Paper

Authors
Hassan Jafarian Dehkordi 1
Gholamreza Rezaeezadeh 2
Masoumeh Bibak 3
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
10.22103/jmmr.2023.20941.1395
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.
Keywords
p-nilpotent
p-supersoluble
weakly S-semipermutable
Subjects
[1] H. G. Bray, M. Weinstein, Between nilpotent and solvable, Polygonal Publishing House, 1982.
[2] W. Guo, The Theory of Classes of Groups, Science Press-Kluwer Academic Publishers, New York, 2000.
[3] B. Huppert, Endliche Gruppen I, Springer, Berlin, New York, 1967.
[4] Isaacs, I. M. (2014). Semipermutable π-subgroups. Arch. Math, 102, 1-5. https://doi.org/10.1007/s00013-013-0604-2
[5] Jafarian Dehkordy, H., & Rezaeezadeh, G. R. (2023). Weakly S-semipermutable sub-groups and p-nilpotency of Groups, Rend. Sem. Mat. Univ. Padova, 150, 137–148. https://doi.org/10.4171/RSMUP/112
[6] Kegel, O. H. (1962). Sylow-Gruppen und Subnormalteiler endlicher Gruppen, Math. Z, 78, 205-221. https://doi.org/10.1007/BF01195169
[7] Kong, Q., & Guo, X. (2018). On weakly s-semipermutable or ss-quasinormal subgroups of finite groups, Ricerche mat, 68, 571-579. https://doi.org/10.1007/S11587-018-0427-3
[8] Li, C., & Yi, X. (2013). Finite groups with some F-supplemented subgroups II, Arab. J. Math. 2, 51-56. https://doi.org/10.1007/s40065-012-0036-z
[9] Li, Y., Qiao, S., Su, N., & Wang, t Y. (2012). On Weakly S-semipermutable subgroups of finite group, J. Algebra, 371, 250-261. https://doi.org/10.1016/j.jalgebra.2012.06.025
[10] Maier, R., & Schmid, P. (1973). The embedding of quasinormal subgroups in finite groups, Math. Z, 131, 269-272. https://doi.org/10.1007/BF01187244
[11] Mirdamadi, S. E., & Rezaeezadeh, G. R. (2016). On finite groups in which SS-semi-permutability is a transitive relation, Journal of Algebraic Systems, 4(1), 29-36. https://doi.org/10.22044/jas.2016.726
[12] Ore, O. (1939). Contributions to the theory of groups of finite order, Duke Math. J, 5(2), 431-460. https://doi.org/10.1215/S0012-7094-39-00537-5
[13] Wei, H., & Wang, Y. (2007). The c-supplemented property of finite groups, Proc. Edinb. Math, 50(2), 493-508.  https://doi.org/10.1017/S0013091504001385
Journal of Mahani Mathematical Research
Volume 13, Issue 1 - Serial Number 26
November 2023
Pages 457-466

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