Non-singular Equations over Groups I

Let G be a group, t an element distinct from G and r(t) = g(1)t(li) ... g(k)t(lk) is an element of G * < t >, where each g, is an element of G of order greater than 2 and the l(i) are non-zero integers such that l(1) + l(2) + ... + l(k) not equal 0 and vertical bar l(i)vertical bar not equal vertical bar l(j)vertical bar for i not equal j. It is known that if k <= 2, then the natural map from G to the one-relator product < G, t vertical bar r(t)> is injective. In this paper, we prove that the same holds for all k is an element of {4, 5}.