Order of $Z^1(G,A)$, the group of $1$-cocycles, when both $G$ and $A$ are finite

Question

I was reading about first cohomology group $H^1(G,A)$ for a group $G$ acting on an abelian group $A$. As one can see from the definition, $H^1(G,A)=Z^1(G,A)/B^1(G,A)$ where $Z^1(G,A)$ is the group of $1$-cocycles. I am actually curious about the order of the group $Z^1(G,A)$. My question is

How the order of $Z^1(G,A)$ depends on order of $G$ and $A$ when both $G$ and $A$ are finite?

I am not familiar with many results about this topic. I am really sorry. Any help about the question will be greatly appreciated. If there is any reference/result even for special class of groups, say $p$-groups, I will be interested to know also.

Thanks.

Answer 1

Suppose that $G$ and $A$ have coprime order. Then $H^1(G,A)$ is trivial, so has order $1$. For a proof, see my notes, Corollary $2.4.5$. This gives many examples.