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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?

Nicolas Boerger
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    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? https://mathoverflow.net/questions/52552/nontrivial-finite-group-with-trivial-group-homologies – Chris Gerig Jan 30 '18 at 22:39
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    Higman's group is not simple. It is SQ-universal: https://mathoverflow.net/questions/221091/properties-of-higmans-group –  Jan 31 '18 at 00:08
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    See Johannes Ebert answer to https://mathoverflow.net/questions/64688/ for a reason why these cannot exist. – Oscar Randal-Williams Jan 31 '18 at 07:56

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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.

Danny Ruberman
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