On one question of A.~N.~Skiba in the theory of $\sigma$-properties of finite groups

I. N. Safonova

Problemy fiziki, matematiki i tehniki, no. 1 (2022), pp. 78-83.

Résumé

All considered groups are finite. Let $G$ be a group, $\sigma$ some partition of the set of all primes $\mathbb{P}$, i. e. $\sigma=\{\sigma_i\mid i\in I\}$, where $\mathbb{P}=\bigcup_{i\in I}\sigma_i$ and $\sigma_i\cap\sigma_j=\varnothing$ for all $i\ne j$, $\sigma(G)=\{\sigma_i\mid \sigma_i\cap\pi(|G|)\ne\varnothing\}$. A group $G$ is called $\sigma$-primary if $G$ is a $\sigma_i$-group for some $i=i(G)$. We say that $G$ is a $\sigma$-tower group if either $G=1$ or $G$ has a normal series $1=G_0$ such that $G_k/G_{k-1}$ is a $\sigma_i$-group, $\sigma_i\in\sigma(G)$, while $G/G_k$ and $G_{k-1}$ are $\sigma_i$-groups for all $k=1,\dots,n$. A subgroup $A$ of $G$ is said to be $\sigma$-subnormal in $G$ if there is a subgroup chain $A=A_0\leqslant A_1\leqslant\dots\leqslant A_t=G$ such that either $A_{i-1}\trianglelefteq A_i$ or $A_i/(A_{i-1})_{A_i}$ is $\sigma$-primary for all $i=1,\dots,t$. In this article, we prove that a non-identity soluble group $G$ is a $\sigma$-tower group if for each $\sigma_i\in\sigma(G)$, where $|\sigma(G)|=n$ a Hall $\sigma_i$-subgroup of $G$ is supersoluble and every $(n+1)$-maximal subgroups of $G$ is $\sigma$-subnormal in $G$. Thus, we give a positive answer to Question 4.8 in [1] in the class of all soluble groups with supersoluble $\sigma$-Hall subgroups.

Détail

Keywords: finite group, $\sigma$-subnormal subgroup, Sylow tower group, $\sigma$-tower group.
Mots-clés : soluble group

Bibliographie

[1] A.N. Skiba, “On $\sigma$-subnormal and $\sigma$-permutable subgroups of finite groups”, J. Algebra, 436 (2015), 1–16 | DOI | MR | Zbl

[2] A.N. Skiba, “O $\sigma$-svoistvakh konechnykh grupp I”, Problemy fiziki, matematiki i tekhniki, 2014, no. 4 (21), 89–96

[3] J.C. Beidleman, A.N. Skiba, “On $\tau_\sigma$-quasinormal subgroups of finite groups”, J. Group Theory, 20:5 (2017), 955–964 | DOI | MR

[4] J. Huang, B. Hu, X. Wu, “Finite groups all of whose subgroups are $\sigma$-subnormal or $\sigma$-abnormal”, Comm. Algebra, 45:1 (2017), 4542–4549 | DOI | MR | Zbl

[5] W. Guo, A.N. Skiba, “On $\sigma$-supersoluble groups and one generalization of CLT-groups”, J. Algebra, 512 (2018), 92–108 | DOI | MR | Zbl

[6] W. Guo, A.N. Skiba, “Finite groups whose $n$-maximal subgroups are $\sigma$-subnormal”, Science in China. Math., 62:7 (2019), 1355–1372 | DOI | MR | Zbl

[7] X. Yi, S.F. Kamornikov, “Finite groups with $\sigma$-subnormal Schmidt subgroups”, J. Algebra, 560:15 (2020), 181–191 | MR | Zbl

[8] Abd El-Rahman Heliel, M. Al-Shomrani, A. Ballester-Bolinches, “On the $\sigma$-length of maximal subgroups of finite $\sigma$-soluble groups”, Mathematics, 8:12 (2020), 2165 | DOI | MR

[9] A. Ballester-Bolinches et al., “On $\sigma$-subnormality criteria in finite $\sigma$-soluble groups”, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas, 114:94 (2020) | DOI | MR | Zbl

[10] A-Ming Liu et al., “$G$-covering subgroup systems for some classes of $\sigma$-soluble groups”, J. Algebra, 582 (2021), 280–293 | MR

[11] A. Ballester-Bolinches, S.F. Kamornikov, X. Yi, “On $\sigma$-subnormality criteria in finite groups”, J. Pure Appl. Algebra, 226:2 (2022), 106822 | DOI | MR | Zbl

[12] Z. Wang et al., “A generalization of $\sigma$-permutability”, Commun. Math. Stat. (to appear)

[13] A.N. Skiba, “On some results in the theory of finite partially soluble groups”, Commun. Math. Stat., 4:3 (2016), 281–309 | DOI | MR | Zbl

[14] A.E. Spencer, “Maximal non-normal chains in finite groups”, Pacific J. Math., 27 (1968), 167–173 | DOI | MR | Zbl

[15] V.S. Monakhov, Vvedenie v teoriyu konechnykh grupp i ikh klassov, Vysshaya shkola, Minsk, 2003

[16] A.I. Mal'cev, Algebraic Systems, Nauka, M., 1970 | MR | Zbl

[17] K. Doerk, T. Hawkes, Finite Soluble Groups, Walter de Gruyter, Berlin–New York, 1992 | MR