Cohomology of finite groups with finite coefficients

Question

I'm wondering if the group cohomology of a finite group $G$ can be made nontrivial with a nice choice of a finite $G$-module M. In other words, given a finite group $G$ and a number $n$, does there exist a finite $G$-module $M$ such that $H^n(G,M)$ is non-zero?

I would also be interested in the special case that $G$ is a finite $p$-group and n = 2. Can I always get $H^2(G,M) \ne 0$ for some finite $M$?

Thanks for your help.

Answer 1

Yes, for each $n\ge 0$ there is a $G$-module $M$ (depending on $n$) such that $H^n(G,M)\neq 0$ (provided $G\neq 1$ finite). .

Such an $M$ can be constructed by induction:

1. First note that $H^i(G,F)=0$ for each free $\mathbb{Z}G$-module $F$ and all $i>0$ by Shapiro's lemma and Brown, VIII, 5.2.

2. Next, show $H^1(G,I_G)=\mathbb{Z}/|G|$ where $I_G \trianglelefteq \mathbb{Z}G$ is the augmentation ideal (and $G$-action is given by multiplication in $\mathbb{Z}G$).

3. Let $n\ge 2$ and suppose $N$ is a $G$-module such that $H^{n-1}(G,N) \neq 0$. Choose a short exact sequence $0 \to M \to F \to N\to 0$ of $G$-modules with $F$ free. Then, by the long exact sequence in cohomology and 1. we obtain the exact sequence $$0=H^{n-1}(G,F) \to H^{n-1}(G,N) \to H^n(G,M) \to H^n(G,F)=0,$$ i.e. $H^n(G,M)\cong H^{n-1}(G,N)\neq 0$.

Note: By starting with $I_G$ you can even arrange $H^n(G,M)=\mathbb{Z}/|G|$.

Answer 2

It is well known that, If $G$ and $M$ are finite $p$-groups, $H^2(G,M)=0$ is equivalent to $H^i(G,M)=0$ for all $i \geq 0$.

If one take any $M$ on which $G$ acts trivially, then $H^1(G,M) \cong Hom(G,M)$, which is not trivial unless $M$ or $G$ are trivial.