Can $\mathbb{Z}_8$ be the commutator subgroup of a group?

Question

I was trying to construct examples of groups with small derived subgroups. I get stuck in the construction of finite group with derived subgroup exactly equal to $\mathbb{Z}_8$. Does such group exists?

Related: http://math.stackexchange.com/questions/361582/which-groups-are-derived-subgroups — Seirios, Sep 23 '15 at 16:47

score 5 · Accepted Answer · answered Sep 23 '15 at 05:03

5

If $D_n$ is the dihedral group of symmetries of a regular $n$-gon, $$D_n=\langle r,s\mid r^n=s^2=1, srs^{-1}=r^{-1}\rangle,$$ then it is easy to show that $[D_n,D_n]=\langle r^2\rangle.$

Can you get the cyclic group of order $8$ out of this?

answered Sep 23 '15 at 05:03

Jyrki Lahtonen

140,891

yes. Thanks Jyrki. – Groups Sep 23 '15 at 05:04

Can $\mathbb{Z}_8$ be the commutator subgroup of a group?

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