Let $G$ be a finite group. I'm trying to understand the structure (if exists) of the set of functions $f:G\times G\to\mathbb{Z}_p$ ($p$ is a given prime) satisfying the condition:
$f(a,b)+f(b,c)+f(c,a)=0$
Where $a,b,c\in G$ and can be subjected to conditions if "all the elements of $G$" gives boring results (especially a condition like "$abc\in S$" for some subset $S\subseteq G$ is good for me).
Also, the most interesting thing for me are such functions that also satisfy the additional constraint $f(0,0)=1$.
Note that a basic structure always exists: such a function $f$ is given by a solution of a set of linear equations of the form $X_{a,b}+X_{b,c}+X_{c,a}=0$, the variables corresponding to the values of $f$. However, I'm afraid this does not give me sufficient insight.
A guiding example for me is group cohomology; the setting is almost the same but the equation defining 2-cocycles is slightly different ($f(a,bc)+f(b,c)-f(ab,c)-f(a,b)=0$ if I'm not mistaken)