Characterizations of $\sigma$-Solvable Finite Groups

All the groups considered in this paper are finite, and $G$ always denotes a finite group; $\sigma$ is a partition of the set $\mathbb{P}$ of all primes, 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 $i \ne j$. A group $G$ is said to be $\sigma$-primary if $G$ is a $\sigma_{i}$-group for some $i=i(G)$, and $\sigma$-solvable if every chief factor of $G$ is $\sigma$-primary. A set of subgroups $\mathcal{H}$ of a group $G$ is called a complete Hall $\sigma$-set of $G$ if every element $\ne 1$ of the set $\mathcal{H}$ is a Hall $\sigma_{i}$-subgroup $G$ for some $i$, and $\mathcal{H}$ contains exactly one Hall $\sigma_{i}$-subgroup of the group $G$ for all $i$ such that $\sigma_{i}\cap \pi(G)\ne \varnothing$. A subgroup $A$ of a group $G$ is said to be $K$-$\mathfrak{S}_{\sigma}$-subnormal in $G$ if $G$ contains a series of subgroups $A=A_{0} \le A_{1} \le\cdots\le A_{t}=G$ such that either $A_{i-1} \trianglelefteq A_{i}$ or the group $A_{i}/(A_{i-1})_{A_{i}}$ is $\sigma$-solvable for all $i=1,\dots,t$. We say that a subgroup $A$ of a group $G$ is weakly $K$-$\mathfrak{S}_{\sigma}$-subnormal in $G$ if $G$ contains $K$-$\mathfrak{S}_{\sigma}$-subnormal subgroups $S$ and $T$ such that $G=AT$ and $A \cap T \le S \le A$. In the present paper, we study conditions under which a group is $\sigma$-solvable. In particular, we prove that a group $G$ is $\sigma$-solvable if and only if at least one of the following two conditions is satisfied: (i) $G$ has a complete Hall $\sigma$-set $\mathcal H$ all of whose elements are weakly $K$-$\mathfrak{S}_{\sigma}$-subnormal in $G$; (ii) in every maximal chain of subgroups $\cdots < M_{3} < M_{2} < M_{1} < M_{0}=G$ of the groups $G$, at least one of the subgroups $M_{3}$, $M_{2}$, or $M_{1}$ is weakly $K$-$\mathfrak{S}_{\sigma}$-subnormal in $G$.