On some applications of Fitting like subgroups of finite groups

TL;DR

This paper explores the properties of groups with specific subgroup normality conditions related to Fitting-like subgroups, providing new characterizations of $\sigma$-nilpotent groups and their hypercenters.

Contribution

It introduces a novel characterization of $\sigma$-nilpotent groups based on $K$-$rak{F}$-subnormality of Sylow subgroups and their relation to Fitting-like subgroups.

Findings

Characterization of $\sigma$-nilpotent groups via $K$-$rak{F}$-subnormal Sylow subgroups

Identification of the $rak{F}$-hypercenter with the largest subgroup normalizing Sylow subgroups

Equivalence of certain subgroup properties with $rak{F}$ being the class of $\sigma$-nilpotent groups

Abstract

In this paper we study the groups all whose maximal or all Sylow subgroups are $K$-$\mathfrak{F}$-subnormal in their product the with generalizations of the Fitting subgroup $\mathrm{F}^*(G)$ and $\mathrm{\tilde F}(G)$. We prove that a hereditary formation $\mathfrak{F}$ contains every group all whose Sylow subgroups are $K$-$\mathfrak{F}$-subnormal in their product with $\mathrm{F}^*(G)$ if and only if $\mathfrak{F}$ is the class of all $\sigma$-nilpotent groups for some partition $\sigma$ of the set of all primes. We obtain a new characterization of the $\sigma$-nilpotent hypercenter, i.e. the $\mathfrak{F}$-hypercenter and the normal largest subgroup which $K$-$\mathfrak{F}$-subnormalize all Sylow subgroups coincide if and only if $\mathfrak{F}$ is the class of all $\sigma$-nilpotent groups.