TITLE: Acyclic Finite Groups
QUESTION [10 upvotes]: 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?

REPLY [21 votes]: 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.