1. All groups in this paper will be assumed to be abelian groups with addition as the group operation. An associative operation on a group G satisfying both distributive laws will be called a multiplication on G. The multiplication gh=O for all g, h C G is called the zero multiplication, and can be defined on every group G. If G allows no other multiplication, then G is called a nil-group. The notion of a nilgroup was introduced by SZELE [2], and studied further in [1]. SZELE proved [1], [2] that a torsion group is nil iff it is divisible and that there are no nil mixed groups. In [3] SZELE generalized the concept of a nil-group as follows. Let n be a positive integer. If there exists a multiplication on G under which Gn#0, but G n+1-=0 under every multiplication, then G is said to have nilstufe n, denoted by v(G)=n, Clearly the nil-groups are the groups G with v(G)= 1. If v(G)>n for every positive integer n, then we will denote v(C)= co. The original intent of the research which led to this paper was to consider a group G = G 1 @G2, v(G1)=v(G2)=l , and to see what could be said about v(G). Szele's theorem assures us that neither G~ nor G 2 can be mixed. We further have that if G~ and G2 are both torsion groups then they are both divisible and G is therefore a divisible, torsion group, and hence nil by Szele's theorem. In theorem 1 we consider the case where G1 is a torsion group, and G 2 is torsion free. However, we consider there the more general situation, v(G1)=n, v(G2)=m, n, m arbitrary positive integers. In w 3 we study the case where both G1 and G 2 are torsion free and of rank 1. If either G1 or G 2 is not nil then its nilstufe is ~, ([1], p. 270). Therefore v(G~)=v(G2)=l is the most general case to consider when both G 1 and G 2 are torsion free groups of rank 1. Tbe case G~ and G2 torsion free and of arbitrary rank remains an open question. 2. THEOREM 1. Let G = Ga | G 2, G~ torsion ,v(Gt)=n, G 2 torsion free, v(G2)=m, then v(G) <= (n+l)(m+l)--l. PROOF. Consider the cartesian product G~XGj, i=1 or 2, j=l or 2. Every multiplication on G is a bilinear mapping of Gi X Gj into G and therefore factors through G~| Gj. If either i= 1 or j= 1 then G~| Gi is a torsion group, and g~gj C G~, gi ~ Gi, gj C G i. We therefore have: