Test Ranks of Free Nilpotent Groups

The test rank tr(G) of G is the minimum cardinality of a test set. In this paper we prove: I. Let N = N rc be a free nilpotent group of rank r ≥ 2 and class c ≥ 2. Then (i) tr(N) = 2 for r odd and c = 2; (ii) tr(N) = 1 in all other cases; (iii) an element g ∈ N 2q, 2 is a test element if and only if it can be written as g = ⋅ s , where t 1,…, t q are nonzero integers. II. Let F r be a free group of rank r ≥ 2 and A n be a free abelian group of rank n ≥ 1. Then the group G = G rn = F r ×A n has the test rank n.