= Abelianization of an odd dihedral group
{title2=$D_{2n}^{\mathrm{ab}}\cong C_2$}
For odd $n\geq3$, the <dihedral group> $D_{2n}$ of order $2n$ has rotation generator $r$ and reflection generator $s$, with $srs=r^{-1}$. The <group commutator> $[r,s]=r^{-2}$ generates the rotation <subgroup>, while the quotient by that <subgroup> is $C_2$. Hence the <commutator subgroup> is exactly $\langle r\rangle$ and the <abelianization> is $C_2$. The only <abelian quotients> are $C_2$ and the trivial group.
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