I'm very familiar with cyclic groups, but I can't seem to understand what a non-cyclic group is like... is there a simple example of one?
(edited... I had no idea that non-cyclic != acyclic)
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In the sense of algebraic topology (group whose classifying space is acyclic), or are you just looking for groups that aren't cyclic...? – Alex Provost Jun 22 '17 at 14:53
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@Jason S A noncyclic group is not the same as an acyclic group, see the link in my answer. – Dietrich Burde Jun 22 '17 at 14:53
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just an example of groups that are not cyclic – Jason S Jun 22 '17 at 14:54
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@DietrichBurde Your concern has led to the askers correction of what the intended question is asking. – amWhy Jun 22 '17 at 14:55
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@amWhy Yes, you are right. Then my answer is no longer helpful. – Dietrich Burde Jun 22 '17 at 14:56
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One of the first examples is Higman's four-generator four-relator group [Hi] $$ \langle x_0,x_1,x_2,x_3 \mid x_{i+1}x_ix_{i+1}^{-1}=x_i^2, i∈ \mathbb{Z}/4\rangle. $$ This group is acyclic, i.e., it has the same constant coefficient homology as the trivial group. For details see here.
Edit: It seems that you wanted an example of a non-cyclic group. Just take $C_2\times C_2$, which cannot have an element of order $4$, hence is not cyclic.
Dietrich Burde
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Given the edit you made to address the earlier edit of the asker, your answer still has merit; no need to delete. – amWhy Jun 22 '17 at 15:00