A polycyclic presentation for the q-tensor square of a polycyclic group

Let G be a group and q a non-negative integer. We denote by v(q)(G) a certain extension of the q-tensor square G circle times(q) G by G x G. In this paper, we describe an algorithm for deriving a polycyclic presentation for G circle times(q) G when G is polycyclic, via its embedding into v(q) (G). Furthermore, we derive polycyclic presentations for the q-exterior square G boolean AND(q) G and for the second homology group H-2 (G, Z(q)). Additionally, we establish a criterion for computing the q-exterior center Z(q)(boolean AND) A(G) of a polycyclic group G, which is helpful for deciding whether or not G is capable modulo q. These results extend to all q >= 0 generalizing methods due to Eick and Nickel for the case q = 0.