Some Structural And Closure Properties Of An Extension Of The Q-Tensor Product Of Groups, Q >= 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 circle times H-q, where G and H are normal subgroups of some group 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 circle times D-q(n) for q odd, where D-n denotes the dihedral group of order 2n. Upper bounds for the exponent of G circle times(q)G are also established for nilpotent groups G of class 3, which extend to all q = 0 similar bounds found by Moravec, P. (2008). The exponents of nonabelian tensor products of groups. J. Pure Appl. Algebra. 212(7):1840-1848.