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I am having troubles understanding the superscript "G/N" in the third term of the standard inflation-restriction exact sequence $$ 0\to H^1(G/N,A^N)\to H^1(G,A)\to H^1(N,A)^{G/N}\to H^2(G/N,A^N)\to H^2(G,A) $$

Is it assumed here that $G/N$ somehow acts on the first cohomology group $H^1(N,A)$ so one could consider the fixed points of this action? If so, the action is quite nonobvious.

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  • See Benson - Representations and Cohomology II section 3.5 for a description of how this action is defined. Briefly, you take a $kG$-projective resolution of $k$ which is a $kN$-projective resolution by restriction, so can be used to calculate $H^1(N,A|_N)$. Then the space of $N$-homs from this resolution to $A$ is a $G$-module ($(g\cdot f)(q)=gf(g^{-1}q)$), but $N$ acts trivially, hence it becomes a $G/N$-module. This induces an action on the Ext-groups. – Matthew Towers Jul 30 '15 at 11:54

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