$\DeclareMathOperator\cd{cd}$Recall that the **rational cohomological dimension** of a group $G$ is the supremum of the set of integers $k$ such that there exists a $\mathbb{Q}[G]$-module $M$ with $H^k(G;M) \neq 0$. Denote this by $\cd_{\mathbb{Q}}(G)$.

If $G$ is a finite group, then it is easy to see that $\cd_{\mathbb{Q}}(G) = 0$.

However, it is not clear to me what the rational cohomological dimension of a locally finite group is (where "locally finite" means that all finitely generated subgroups are finite). Since homology commutes with direct limits, the rational **homological** dimension of these group are $0$.

So my question is what can be said about the rational cohomological dimension of a locally finite group $G$. I mostly care about countable $G$. In fact, the easiest example where I don't know the answer is $$G = \bigoplus_{k=1}^{\infty} \mathbb{Z}/2\mathbb{Z}.$$