Some Structural and Closure Properties of an Extension of the $q$-tensor Product of Groups, $q\ge 0$

In this work we study some structural properties of the group ${\eta }^q(G,H)$, $q$ a non-negative integer, which is an extension of the $q$-tensor product $G \otimes^q H)$,where$G$and$H$arenormalsubgroupsofsomegroup$L$. We establish by simple arguments some closure properties of ${\eta }^q(G,H)$ when $G$ and $H$ belong to certain Schur classes. This extends similar results concerning the case $q=0$ found in the literature. Restricting our considerations to the case $G=H$, we compute the $q$-tensor square $D_n \otimes^q D_n$for$q$odd,where$D_n$denotesthedihedralgroupoforder$2n$\mathrm{.}Upperboundstotheexponentof$G \otimes^q G$ are also established for nilpotent groups $G$ of class $\le 3$, which extend to all $q\ge 0$ similar bound found by Moravec in [21].