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Let $G$ be a group, and let $A$ be a $G$-module. Then for every subgroup $H$ of $G$, $A$ is also an $H$-module. Furthermore, there's a map $H^2(G,A)\rightarrow H^2(H,A)$.

I would like to know something about those subgroups that satisfy that the image of this map is trivial. In particular I wonder if there is a maximal such $H$.

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    If your group finite?2012-01-23
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    No, not necessarily. Although in the cases that I'm thinking of A is finite.2012-01-23
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    Just out of curiosity, what would you have said under the assumption that $G$ is finite?2012-01-23
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    For the finite case, there is no unique maximal such $H$; just consider $A_5$ with trivial module over $\mathbb{F}_2$, and think about $H$ being a Sylow 3-group or 5-group.2012-01-23
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    Yeah, that was where I was going :) In the infinite case: I think there are non-free groups such that every proper subgroup is free.2012-01-23

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