This article is cited in 7 papers

MATHEMATICS

On $\sigma$-properties of finite groups II

A. N. Skiba

F. Scorina Gomel State University, Gomel, Belarus

Abstract: Let $G$ be a finite group, $\sigma=\{\sigma_i \mid i\in I\}$ some partition of the set $\mathbb{P}$ of all primes and $\Pi$ a subset of the set $\sigma$. A set $\mathcal{H}$ of subgroups of $G$ is said to be a complete Hall $\Pi$-set of $G$ if $\mathcal{H}$ contains exact one Hall $\sigma_i$-subgroup of $G$ for every $\sigma_i\in\Pi$ such that $\sigma_i\cap\pi(G)\ne\varnothing$. We say also that $G$ is: $\Pi$-full if $G$ possess a complete Hall $\Pi$-set; a $\Pi$-full group of Sylow type if for each $\sigma_i\in\Pi$, every subgroup $E$ of $G$ is a $D_{\sigma_i}$-group, that is, $E$ has a Hall $\sigma_i$-subgroup $H$ and every $\sigma_i$-subgroup of $E$ is contained in some conjugate of $H^x$ ($x\in E$). In this paper we study properties of finite $\Pi$-full groups. The work continues the research of the paper [1].

Keywords: finite group, $\Pi$-full group, $\sigma$-soluble group, $\sigma$-nilpotent group, $\sigma$-quasinilpotent group.

UDC: 512.542

Received: 14.07.2015