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If you know a bit of Algebraic Topology (in particular the dreaded spectral sequences), the following is a nice way to see this. The well known Hopf Theorem states that for $G$ a compact connected Li …

If $G$ is a compact connected topological group and $A$ is a locally compact abelian topological group, then for any map $f:BG\to BA$ the looped map $\Omega f:\Omega BG\to \Omega BA$ is homotopically …

If $X$ is a space of finite type (meaning that the homology groups $H_i(X)$ are all finitely generated, a condition which applies in particular to $X=BG$ for $G$ a compact Lie group) then for each $n$ …

Any finitely presented group of type FL admits a finite classifying space. (A group $G$ is of type FL if $\mathbb{Z}$ admits a finite length resolution by finitely generated, free $\mathbb{Z}G$-module …

In the paper Arkowitz, Martin Categories equivalent to the category of rational H-spaces, Manuscripta Math. 64 (1989), no. 4, 419–429 it is shown that the rational homotopy equivalence $G_\mathbb{Q} …

In fact "$\mathbb{Q}$-acyclic" means "having the same rational homology groups as a point". In particular, the classifying space of any finite group is $\mathbb{Q}$-acyclic, as can be seen by a simple …