Let $G$ be any group and $M$ any $G-$module. Suppose that $N$ is a $G$-submodule of $M$. Is there a relation between the cohomology groups $H^{i}(G,N)$ and $H^{i}(G,M)$?
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https://en.wikipedia.org/wiki/Group_cohomology#Long_exact_sequence_of_cohomology – hunter Jun 16 '21 at 22:34
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A $G$-submodule $N$ of $M$ is a $G$-monomorphism $N \to M$. The cohomology groups are functors, so in all dimensions they induce maps $H^q(G,N)\to H^q(G,M)$. From the long exact sequence in cohomology we also have maps $H^q(G,M)\to H^q(G,M/N) \to H^{q+1}(G,N)$ coming from the short exact sequence $0 \to N \to M \to M/N \to 0$. All of this is found in the Wikipedia article linked by hunter. – k-rational Jun 26 '21 at 16:41