On NH-embedded and SS-quasinormal subgroups of finite groups

Year 2024, , 121 - 129, 09.01.2024

Weicheng Zheng Liang Cuı Wei Meng Jiakuan Lu

https://doi.org/10.24330/ieja.1299719

Abstract

Let $G$ be a finite group. A subgroup $H$ is called $S$-semipermutable in $G$ if $HG_p$ = $G_pH$ for any $G_p\in Syl_p(G)$ with $(|H|, p) = 1$, where $p$ is a prime number divisible $|G|$. Furthermore, $H$ is said to be $NH$-embedded in $G$
if there exists a normal subgroup $T$ of $G$ such that $HT$ is a Hall subgroup of $G$ and
$H \cap T \leq H_{\overline{s}G}$, where $H_{\overline{s}G}$ is the largest $S$-semipermutable subgroup of $G$ contained in
$H$, and $H$ is said to be $SS$-quasinormal in $G$ provided there is a supplement $B$ of $H$ to $G$ such that $H$ permutes with every Sylow subgroup of $B$. In this paper, we obtain some criteria for $p$-nilpotency and Supersolvability of a finite
group and extend some known results concerning $NH$-embedded and $SS$-quasinormal subgroups.

Keywords

$SS$-quasinormal, $NH$-embedded, $p$-nilpotent group, supersolvable

References

• M. Asaad and A. Heliel, On S-quasinormally embedded subgroups of finite groups, J. Pure Appl. Algebra, 165(2) (2001), 129-135.
• Z. Chen, On a theorem of Srinivasan, J. Southwest Normal Univ. Nat. Sci. (Chinese), 12(1) (1987), 1-4.
• W. E. Deskins, On quasinormal subgroups of finite groups, Math. Z., 82 (1963), 125-132.
• K. Doerk and T. Hawkes, Finite Soluble Groups, Walter de Gruyter, Berlin, 1992.
• Y. Gao and X. Li, On NH-embedded subgroups of finite groups, J. Algebra Appl., 21(10) (2022), 2250200 (11 pp).
• B. Huppert, Endliche Gruppen, Springer-Verlag, Berlin, 1967.
• I. M. Isaacs, Semipermutable $\pi$-subgroups, Arch. Math. (Basel), 102(1) (2014), 1-6.
• O. H. Kegel, Sylow-Gruppen und Subnormalteiler endlicher Gruppen, Math. Z., 78 (1962), 205-221.
• S. Li, Z. Shen, J. Liu and X. Liu, The influence of SS-quasinormality of some subgroups on the structure of finite groups, J. Algebra, 319(10) (2008), 4275-4287.
• Y. Li, S. Qiao, N. Su and Y. Wang, On weakly s-semipermutable subgroups of finite groups, J. Algebra, 371 (2012), 250-261.
• P. Schmid, Subgroups permutable with all Sylow subgroups, J. Algebra, 207(1) (1998), 285-293.
• Q. Zhang and L. Wang, The influence of s-semipermutable subgroups on the structure of finite groups, Acta Math. Sinica (Chinese Ser.), 48 (2005), 81-88.