A group is called acyclic if its classifying space has the same homology of a point. Examples of acyclic groups include Higman's group with four generators and relations, also known for being an example of an infinite, finitely generated group with no finite quotients, and "SQ-Universal": 
$$\langle x\_0, x\_1,x\_2,x\_3\mid x\_{i+1}x\_i x\_{i+1}= x\_i^2 \hskip .1 in \mathrm{for}\hskip .1 in i=0,\ldots, 3\rangle$$
and the group of bijections of an infinite countable set. 


Is there an example of a finite acyiclic group? Or a reason why such a group must be infinite? 



Acyclic Finite Groups
An acyclic finite group is trivial. In fact something even stronger is true. See Culler, Marc Homology equivalent finite groups are isomorphic. Proc. Amer. Math. Soc. 72 (1978), no. 1, 218–220. 


