On generalized $σ$-soluble groups

Abstract

Let $σ=\{σ_{i} | i\in I\}$ be a partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. Let $σ(G)=\{σ_{i} : σ_{i}\cap π(G)\ne \emptyset$. A set ${\cal H}$ of subgroups of $G$ is said to be a complete Hall $σ$-set of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $σ_{i}$-subgroup of $G$ for some $i\in I$ and $\cal H$ contains exactly one Hall $σ_{i}$-subgroup of $G$ for every $i$ such that $σ_{i}\in σ(G)$. We say that $G$ is $σ$-full if $G$ possesses a complete Hall $σ$-set. A complete Hall $σ$-set $\cal H$ of $G$ is said to be a $σ$-basis of $G$ if every two subgroups $A, B \in\cal H$ are permutable, that is, $AB=BA$. In this paper, we study properties of finite groups having a $σ$-basis. In particular, we prove that if $G$ has a a $σ$-basis, then $G$ is generalized $σ$-soluble, that is, $G$ has a complete Hall $σ$-set and for every chief factor $H/K$ of $G$ we have $|σ(H/K)|\leq 2$. Moreover, answering to Problem 8.28 in [A.N. Skiba, On some results in the theory of finite partially soluble groups, Commun. Math. Stat., 4(3) (2016), 281--309], we prove the following Theorem A. Suppose that $G$ is $σ$-full. Then every complete Hall $σ$-set of $G$ forms a $σ$-basis of $G$ if and only if $G$ is generalized $σ$-soluble and for the automorphism group $G/C_{G}(H/K)$, induced by $G$ on any its chief factor $H/K$, we have either $σ(H/K)=σ(G/C_{G}(H/K))$ or $σ(H/K) =\{σ_{i}\}$ and $G/C_{G}(H/K)$ is a $σ_{i} \cup σ_{j}$-group for some $i\ne j$.