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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$\sigma_{i}\cap \sigma_{j}=\emptyset$ for all $i\neq j$. Let $G$ be a finitegroup. 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 saidto 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 normalseries $1 = G_1<G_2<\cdots< G_t< G_{t+1} = G$ and a complete Hall $\sigma$-set$\{H_{1}, H_{2}, \cdots, H_{t}\}$ such that $G_iH_i = G_{i+1}$ for all $i=1,2,\ldots t$. In this paper, we give a characterizations of$\sigma$-dispersive group, which give a positive answer to an open problem ofSkiba in the paper.

On an open problem of Skiba