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

Abstract

In the present paper, the authors extend to the [Formula: see text]-tensor square [Formula: see text] of a group [Formula: see text], [Formula: see text] an odd positive integer, some structural results due to Blyth, Fumagalli and Morigi concerning the non-abelian tensor square [Formula: see text] ([Formula: see text]). The results are applied to the computation of [Formula: see text] for finitely generated nilpotent groups [Formula: see text], specially for free nilpotent groups of finite rank. We also generalize to all [Formula: see text] results of Bacon regarding an upper bound to the minimal number of generators of the non-abelian tensor square [Formula: see text] when [Formula: see text] is a [Formula: see text]-generator nilpotent group of class 2. We end by computing the [Formula: see text]-tensor squares of the free [Formula: see text]-generator nilpotent group of class 2, [Formula: see text]. This shows that the above mentioned upper bound is also achieved for these groups when [Formula: see text] odd.