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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)

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    I have a doubt that others may as well have: $f: G \to \mathbb Z_p$ . So, what is the meaning of $f(a,b)$. Do you mean that $f$ is function on the set product?2012-02-05
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    Yes, I meant $f:G\times G\to \mathbb{Z}_p$. Thank you.2012-02-05
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    I think, $f(a,b)=ab-ba$ is a good map to study and understand.2012-02-05
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    Did you mean $f(e,e)=0$ instead of $f(0,0)=1$? Assuming your $0$ denotes the identity of $G$, the condition $f(0,0)=1$ can only be satisfied if $p=3$.2012-02-05
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    Also note that your conditions do not involve the group structure of $G$ at all. You probably want to require a bit more than you did.2012-02-05
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    Marc, note that I allow you to limit the triples $a,b,c$ involved to those having the property $abc\in S$ for a given $S\subseteq G$. This solves the $f(0,0)=1$ problem if $0\notin S$, and involves the group operation on $G$.2012-02-05
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    @GadiA: But note that what you allow is _weakening_ the condition you impose; for an extreme case if $S=\emptyset$ you are no longer requiring anything for $f$. You cannot expect to find _more_ structure by weakening the constraint on $f$.2012-02-05
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    Marc, why not? A group has less conditions than a field, but no one will say they do not have an interesting structure - quite different than that of fields. Of course I don't want to drop ALL demands; in the setting the problem arose in I have a specific S of size 3.2012-02-05

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