On L.G. Kovàcs' problem

Abstract

"Kourovka notebook" contains the question due to L.G. Kovàcs (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 $\mathfrak{X}$.