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I wonder if the group cohomology of a finite group $G$ with coefficients in $\mathbb{Z}$ is finite. This statement may be too strong. I am interested in, for instance, dihedral group. $ G=D_{2n}=\langle a,b | \ a^n=b^2=abab=e \rangle $ Assume that $a$ acts trivially and $b$ acts as $-id$ on $\mathbb{Z}$.

First cohomology is $\mathbb{Z}^G=0$. The second cohomology already seems quite involved to me.

I read several post about group cohomology on StackExchange and MathOverflow, but I still have trouble computing explicit example and getting intuition behind the concept.

Thank you for your assistance.

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    I will check it up. Thanks, Ralph.2012-08-11

1 Answers 1

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The answer is yes, since it's torsion (killed by the order of $G$) and finitely generated (since you can pick a resolution by finitely generated abelian groups).

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    OK. I will take a look at the reference. Thanks, Ralph.2012-08-12