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