The permutability of $σ_i$-sylowizers of some $σ_i$-subgroups in finite groups

Abstract

Let $σ=\{σ_{i}|i\in I\}$ be a partition of the set of all primes $\mathbb{P}$, $G$ a finite group and $σ(G)=\{σ_{i}|σ_{i}\cap π(|G|)\neq\emptyset\}$. A subgroup $S$ of a group $G$ is called a $σ_i$-sylowizer of a $σ_i$-subgroup $R$ in $G$ if $S$ is maximal in $G$ with respect to having $R$ as its Hall $σ_i$-subgroup. The main aim of this paper is to investigate the influence of $σ_i$-sylowizers on the structure of finite groups. We obtained some new characterizations of supersoluble groups by the permutability of the $σ_i$-sylowizers of some $σ_i$-subgroups.