On an open problem of Skiba

Abstract

Let $\sigma =\{{\sigma }_i|i\in I\}$ be some partition of the set $\mathbb{P}$ of all primes, that is, $\mathbb{P}={\bigcup }_{i\in I}{\sigma }_i$ and ${\sigma }_i\cap {\sigma }_j=\varnothing$ for all $i\ne j$. Let $G$ be a finite group. A set $\mathcal{H}$ of subgroups of $G$ is said to be a complete Hall $\sigma$-set of $G$ if every non-identity member of $\mathcal{H}$ is a Hall ${\sigma }_i$-subgroup of $G$ and $\mathcal{H}$ contains exactly one Hall ${\sigma }_i$-subgroup of $G$ for every ${\sigma }_i\in \sigma (G)$. $G$ is said to be a $\sigma$-group if it possesses a complete Hall $\sigma$-set. A $\sigma$-group $G$ is said to be $\sigma$-dispersive provided $G$ has a normal series $1=G_1<G_2<\cdots <G_t<G_{t+1}=G$ and a complete Hall $\sigma$-set $\{H_1,H_2,\cdots ,H_t\}$ such that $G_i{H}_i=G_{i+1}$ for all $i=1,2,\dots t$. In this paper, we give a characterizations of $\sigma$-dispersive group, which give a positive answer to an open problem of Skiba in the paper.