On L.G. Kov\`acs' problem

TL;DR

This paper proves that a dihedral group of order 18, when appearing as a section in a product of locally finite groups, must be a section of one of the factors, extending to certain semidirect products in locally finite varieties.

Contribution

It provides a positive answer to Kov extquotesingle acs' problem for locally finite groups and generalizes to semidirect products of cyclic p-groups with groups of prime order.

Findings

Dihedral group of order 18 is a section of a locally finite product only if it appears in one factor.

Semidirect products of cyclic p-groups with groups of prime order are sections of groups in the same locally finite variety.

The result extends Kov extquotesingle acs' problem to a broader class of groups in locally finite varieties.

Abstract

"Kourovka notebook" contains the question due to L.G. Kov\`acs (Problem 8.23): If the dihedral group $D$ of order 18 is a section of a direct product $X\times Y$, must at least one of $X$ and $Y$ have a section isomorphic to $D$? The goal of our short paper is to give the positive answer to this question provided that $X$ and $Y$ are locally finite. In fact, we prove even more: If a non-trivial semidirect product $D=A\rtimes B$ of a cyclic $p$-group $A$ and a group $B$ of order $q$, where $p$ and $q$ are distinct primes, lies in a locally finite variety generated by a set $\mathfrak{X}$ of groups, then $D$ is a section of a group from $\mathfrak{X}$.