This article is cited in 5 papers

Finite groups with subnormal residuals of Sylow normalizers

T. I. Vasilyeva^a, A. G. Koranchuk<sup>b

a</sup> Belarusian State University of Transport
^b Gomel State University named after Francisk Skorina

Abstract: Considering a nonempty formation $\mathfrak{X}$ of nilpotent groups, we prove that a group $G$ is an extension of a nilpotent group by an $\mathfrak{X}$-group if and only if every Sylow normalizer in $G$ is solvable and its $\mathfrak{X}$-residual is subnormal in $G$. We also show that $G$ is supersolvable if and only if every Sylow normalizer in $G$ is supersolvable and its nilpotent residual is subnormal in $G$.

Keywords: finite group, Sylow normalizer, subnormal subgroup, formation, $\mathfrak{X}$-residual, supersolvable group.

UDC: 512.542

MSC: 35R30

Received: 17.12.2021
Revised: 09.02.2022
Accepted: 10.02.2022

DOI: 10.33048/smzh.2022.63.407

 English version: Siberian Mathematical Journal, 2022, 63:4, 670–676