# Acyclic Finite Groups

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?

• There are no nontrivial finite acyclic groups. A result of Richard Swan says that a group with $p$-torsion has nontrivial mod-$p$ cohomology in infinitely many dimensions, hence nontrivial integral homology. But there is a related question that may be relevant to your interests: Is there a finite group with many trivial homologies? mathoverflow.net/questions/52552/… Jan 30, 2018 at 22:39
• Higman's group is not simple. It is SQ-universal: mathoverflow.net/questions/221091/properties-of-higmans-group
– user6976
Jan 31, 2018 at 0:08
• See Johannes Ebert answer to mathoverflow.net/questions/64688 for a reason why these cannot exist. Jan 31, 2018 at 7:56