For a finite cyclic group $G$, there is the Herbrand quotient in the theory of group cohomology. I calculated some of those quotients and I always came up with an Integer as solution. I failed at proving that for the Herbrand quotient $h(M)$ of a $G$-Module $M$ the following is true:
$$h(M)\in\mathbb{Z}\quad\text{for all $G$-Modules $M$}.$$
So my question is if this statement is true, or if there are examples for $G$-Modules $M$ with $h(M)\in\mathbb{Q}-\mathbb{Z}$?
No, there are examples with non-integer Herbrand quotient. Here I give two of them:
1- $G=\mathbb{Z}/n\mathbb{Z}$ and $M$ an $n$-divisible abelian group with finite nontrivial $n$-torsion elements. Then the Herbrand quotient of the trivial action is $\frac{1}{|\{x\in M:\,nx=0\}|}$.
2- Consider the only nontrivial action of $G=\mathbb{Z}/2\mathbb{Z}$ on $M=\mathbb{Z}$, (so the action of the nontrivial element is $x\to -x$). Then $H^2(G,M) = 0, H^1(G,M) = \mathbb{Z}/2\mathbb{Z}$ and $h(M)=\frac12$.
