This is a question that I haven't been able to find answers for in the books I've looked at, so I wanted to ask it here to get some clarification.
Given an abelian group $A$ and a Galois group $G$ which acts on $A$, we can consider the first cohomology group $H^1(G,A)$. My question is: how does $G$ act on the group $H^1(G,A)$? What is the formula for this action?
I'm guessing that the $G$-action is given by first conjugating the input, and then mapping it over to $A$. That is, given an equivalence class $[\sigma] \in H^1(G,A)$ and a group element $g \in G$, define the class $g \cdot [\sigma]$ as the class of the following co-cycle from $G$ to $A$: $$x \mapsto \sigma(gxg^{-1}). $$
Is this correct? I'm sure that there's some book that has this information but I can't find it in the books I've looked at (Silverman's AEC, Serre's Galois Cohomology). If anyone has any info about this, I'd really appreciate it. Thanks for the help!