Is it constant?

Question

Let $G$ be an abelian group, and let $n:G\times G\rightarrow \mathbb{Z}$ be a map satisfying for all $u,v,w\in G$ $$ n(u+v,w)+n(u,v)=n(u,v+w)+ n(v,w)$$ and $n(0,x)=n(y,0)=0$ for all $x,y\in G$

Is $n$ necessarily a constant?

Answer 1

Let $G$ be the cyclic group of order two, generated by $a$ ($2 a=0$). Then define $n(u,v)=0$ if $u$ or $v$ are the identity element & $n(a,a)=1$ (or whatever value you like). There are eight instance of $n(u+v,w)+n(u,v)=n(u,v+w)+ n(v,w)$ to check, and these are all satisfied.