The q-tensor square of finitely generated nilpotent groups, q odd

Abstract

In the present paper the authors extend to the q−tensor square G ⊗ q G of a group G, q a non-negative integer, some structural results due to R. D. Blyth, F. Fumagalli and M. Morigi concerning the non-abelian tensor square G⊗G (q = 0). The results are applied to the computation of G⊗ q G for finitely generated nilpotent groups G, specially for free nilpotent groups of finite rank. We also generalize to all q ≥ 0 results of M. Bacon regarding an upper bound to the minimal number of generators of the non-abelian tensor square G ⊗ G when G is a n−generator nilpotent group of class 2. We end by computing the q−tensor squares of the free n−generator nilpotent group of class 2, n ≥ 2, for all q ≥ 0. This shows that the above mentioned upper bound is also achieved for these groups when q > 1.