On an open problem of Skiba

Abstract

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