On an open problem of Skiba

TL;DR

This paper characterizes $\sigma$-dispersive groups, providing a solution to an open problem posed by Skiba, by analyzing the structure of finite groups with specific Hall subgroup properties.

Contribution

It offers a new characterization of $\sigma$-dispersive groups, advancing understanding of their structure and resolving an open problem by Skiba.

Findings

Provides a characterization of $\sigma$-dispersive groups.

Answers positively to Skiba's open problem.

Enhances understanding of Hall subgroup structures in finite groups.

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 $\sigma_{i}\cap \sigma_{j}=\emptyset$ 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 $\sigma$-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 $G_iH_i = G_{i+1}$ 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 of Skiba in the paper.