Test ranks of free nilpotent groups

摘要

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 is an element of N-2q,N-2 is a test element if and only if it can be written as g = x(1), x(2)(t1)center dot center dot center dotx(2q-1), x(2q)(tq), 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 x A(n) has the test rank n.