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I think this is a pretty simple question, but I cannot seem to find the answer anywhere:

In the context of group cohomology of a group $G$, $G$-invariant maps are often mentioned. What is the definition of $G$-invariant. Also, is it true that maps need to be $G$-invariant between cochains to make the maps carry through to the cohomology groups (I remember reading this somewhere, but I cannot find this either)

Thank you for any help.

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A map $f:X\to Y$ is $G$-invariant if $f(gx)=f(x)$ for all $x\in X, g\in G$.

For a map to carry through to cohomology, it must send coboundaries to coboundaries and cocycles to cocycles. Is that what you mean?

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    No, it depends on the map and the context under which you're working. In my answer above, $X$ can be any $G$-set, so it may not even make sense to talk about group cohomology2012-11-14