Presentations and Cartesian product of $m$-groups

It is shown that any convex $m$-subgroup of a Cartesian product of $m$-groups that admits a faithful $m$-transitive representation is a convex m-subgroup of an appropriate projection of the Cartesian product of $m$-groups. This implies that the Cartesian product of $m$-groups does not admit a faithful $m$-transitive representation.