On some applications of Fitting like subgroups of finite 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 ${\mathrm{F}}^{\ast }(G)$ and $\tilde{\mathrm{F}}(G)$. We prove that a hereditary formation $\mathfrak{F}$ contains every group all whose Sylow subgroups are $K$-$\mathfrak{F}$-subnormal in their product with ${\mathrm{F}}^{\ast }(G)$ 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.

References in corpus (1)

• On partially conjugate-permutable subgroups of finite groups