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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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    do you mean each cohomology group $H^p(G;\mathbb{Z})$ is a finite group or do you mean the whole cohomology ring $H^*(G;\mathbb{Z})$? The latter statement is of course false, seen e.g. in the finite cyclic groups.2012-08-10
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    Sorry for the confusion. I menat that each $H^{p}(G,M)$ is finite.2012-08-10
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    There are projective resolutions for the dihedral group (due to Wall) that can be used to compute the cohomology for every coefficient module. In particular it shouldn't be to hard to figure out $H^2(D_{2n};-)$.2012-08-10
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    I will check it up. Thanks, Ralph.2012-08-11

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