On Factor Groups of some Groups

TL;DR

This paper investigates the properties of factor groups within specific classes of groups defined by finiteness and periodicity conditions, proving that certain quotient groups remain within these classes under specified conditions.

Contribution

It establishes that factor groups of $p$-biprimitively finite and periodic $p$-conjugatively biprimitively finite groups retain their class membership when quotiented by particular periodic subgroups.

Findings

Factor groups preserve class membership under certain conditions.

Periodic subgroups with specific series lead to quotient groups in the same class.

Preliminary results support the main theorems.

Abstract

Let for a prime $p$, $\mathfrak{X}$ (respectively $\mathfrak{Y}$) be the class of all $p$-biprimitively finite (respectively periodic $p$-conjugatively biprimitively finite) groups and $G\in \mathfrak{X}$ (respectively $G\in \mathfrak{Y}$), $V$ be a periodic subgroup of $G$ having an ascending series of normal in $G$ subgroups such that each its factor is an almost layer-finite group or a locally graded group of finite special rank, or a $WF$-group with $min-q$ on all primes $q$. We prove that $G/V \in \mathfrak{X}$ (respectively $G/V\in \mathfrak{Y}$). Also some interesting and useful preliminary results are obtained.