MATHEMATICS

Finite groups with partially $p$-subnormal Schmidt subgroups

I. M. Dergacheva, E. A. Zadorozhnyuk, I. P. Shabalina

Belarusian State University of Transport, Gomel

Abstract: Throughout the paper, all groups are finite and $G$ always denotes a finite group; $G$ is called a Schmidt group if $G$ is not nilpotent, but every proper subgroup of $G$ is nilpotent. A subgroup $A$ of $G$ is called $\mathfrak{U}_p$-normal in $G$ if every principal $pd$ factor of $G$ between $A_G$ and $A^G$ is cyclic. We say that a subgroup $A$ of $G$ is partially $p$-subnormal in $G$ if $A=\langle L, T\rangle$ for some subnormal subgroup $L$ and $\mathfrak{U}_p$-normal subgroup $T$ of $G$. In this paper, we prove the following theorem.
{\bf Theorem}. If every Schmidt subgroup of a group $G$ is partially $p$-subnormal in $G$, then the derived subgroup $G'$ of $G$ is $p$-nilpotent.

Keywords: finite group, Schmidt group, $p$-nilpotent group, $\mathfrak{U}_p$-normal subgroup, partially $p$-subnormal subgroup.

UDC: 512.542

Received: 04.03.2025

DOI: 10.54341/20778708_2025_2_63_51