If $G \neq \lbrace 1 \rbrace$ is a finite group with classifying space $BG$ 
then there are infinitely many i such that $H^i(BG,\mathbb{Z}) \neq 0$. This 
can be found, for example, there:  

[https://mathoverflow.net/questions/64688/non-vanishing-of-group-cohomology-in-sufficiently-high-degree/64702#64702][1]

As a consequence, the CW-complex $BG$ (unique up to homotopy) can not be of finite dimension. 

Question: Are there alternative proofs for this observation. In particular, I would be interested in knowing if there is a purely topological proof without homological algebra.


  [1]: https://mathoverflow.net/questions/64688/non-vanishing-of-group-cohomology-in-sufficiently-high-degree/64702#64702