TITLE: Can $\mathbb{Z}_8$ be the commutator subgroup of a group?
QUESTION [6 upvotes]: 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?

REPLY [5 votes]: 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?