On $\sigma$-$c$-subnormal subgroups of finite groups

Abstract

Let $ \sigma=\{\sigma_i:i\in I\} $ be a partition of the set $ \mathbb{P} $ of all primes. A finite group $ G $ is called $ \sigma $-primary if the prime divisors of $|G|$, if any, all belong to the same member of $ \sigma $. A finite group $ G $ is called $ \sigma $-soluble if every chief factor of $ G $ is $ \sigma$-primary. A subgroup $H$ of a group $G$ is called $\sigma$-subnormal in $G$ if there is a chain of subgroups $H=H_0\leq H_1\leq\cdots\leq H_n=G$ such that either $ H_{i-1} $ is normal in $ H_i $ or $ H_{i}/(H_{i-1})_{H_{i}} $ is $ \sigma $-primary for all $ i=1,\dots,n $; A subgroup $H$ of a group $G$ is called $\sigma$-$c$-subnormal in $G$ if there is a subnormal subgroup $T$ of $G$ such that $G=HT$ and $H\cap T\leq H_{\sigma G}$, where the subgroup $H_{\sigma G}$ is generated by all $\sigma$ subnormal subgroups of $G$ contained in $H$. In this paper, we investigate the influence of $\sigma$-$c$-subnormality of some kinds of maximal subgroups on $\sigma$-solubility of finite groups, which generalizes some known results.

Keywords

• finite group
• finite $\sigma$-soluble group
• $ \sigma $-$c$-Subnormal subgroup

References

1. [1] A. Ballester-Bolinches and M.C. Pedraza-Aguilera, On minimal subgroups of finite groups, Acta Math. Hungar. 73, 335342, 1996.
2. [2] J.D. Dixon and B. Mortimer, Permutation Groups, Springer, New York, 1996.
3. [3] K. Doerk and T. Hawkes, Finite Soluble Groups, Walter de Gruyter, Berlin, New York, 1992.
4. [4] W. Kimmerle, R. Lyons, R. Sandling and D.N. Teague, Composition factors from the group ring and Artin’s theorem on orders of simple groups, Proc. London Math. Soc.(3), 60 (1), 89-122, 1990.
5. [5] J. Lafuente, Eine Note über nichtabelsche Hauptfaktoren und maximale Untergruppen einer endlichen Gruppe, Comm. Algebra, 13 (9), 2025-2036, 1985.
6. [6] R.M. Peacock, Groups with a cyclic sylow subgroup, J. Algebra, 56 (2), 506-509, 1979.
7. [7] A.N. Skiba, On $\sigma$-subnormal and $\sigma$-permutable subgroups of finite groups, J. Algebra, 436, 1-16, 2015.
8. [8] N. Su, C.C. Cao and S.H. Qiao, A note on maximal subgroups of $\sigma$-soluble groups, Comm. Algebra, 50 (4), 1580-1584, 2022.

9. [9] Y.M. Wang, C-normality of groups and its properties, J. Algebra, 180, 954-965, 1996.