On generalized $\sigma$-soluble groups

Let $\sigma =\{\sigma_{i} | i\in I\}$ be a partition of the set of all primes $\Bbb{P}$ and $G$ a finite group. Let $\sigma (G)=\{\sigma _{i} : \sigma _{i}\cap \pi (G)\ne \emptyset$. A set ${\cal H}$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every member $\ne 1$ of ${\cal H}$ is a Hall $\sigma _{i}$-subgroup of $G$ for some $i\in I$ and $\cal H$ contains exactly one Hall $\sigma _{i}$-subgroup of $G$ for every $i$ such that $\sigma _{i}\in \sigma (G)$. We say that $G$ is $\sigma$-full if $G$ possesses a complete Hall $\sigma$-set. A complete Hall $\sigma$-set $\cal H$ of $G$ is said to be a $\sigma$-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 $\sigma$-basis. In particular, we prove that if $G$ has a a $\sigma$-basis, then $G$ is generalized $\sigma$-soluble, that is, $G$ has a complete Hall $\sigma$-set and for every chief factor $H/K$ of $G$ we have $|\sigma (H/K)|\leq 2$. 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 $\sigma$-full. Then every complete Hall $\sigma$-set of $G$ forms a $\sigma$-basis of $G$ if and only if $G$ is generalized $\sigma$-soluble and for the automorphism group $G/C_{G}(H/K)$, induced by $G$ on any its chief factor $H/K$, we have either $\sigma (H/K)=\sigma (G/C_{G}(H/K))$ or $\sigma (H/K) =\{\sigma _{i}\}$ and $G/C_{G}(H/K)$ is a $\sigma _{i} \cup \sigma _{j}$-group for some $i\ne j$.