Derived subgroup of dihedral group

The derived subgroup (also known as the commutator subgroup) of the dihedral group $D_{2n}$ is the subgroup generated by all the commutators,

$D_{2n}'=\{g^{-1}h^{-1}gh:g,h\in D_{2n}\}$.

It turns out that the derived subgroup of the dihedral group is cyclic (for any $n$). Furthermore, $D_{2n}'=\langle a^{2}\rangle$, where $D_{2n}=\langle x,a\mid a^{n}=x^{2}=e,xax^{-1}=a^{-1}\rangle$.

Table of examples

Dihedral group	Derived subgroup (up to isomorphism)
Dihedral group:D4	Trivial group
Dihedral group:D6	Cyclic group:Z3
Dihedral group:D8	Cyclic group:Z2
Dihedral group:D10	Cyclic group:Z5
Dihedral group:D12	Cyclic group:Z3
Dihedral group:D14	Cyclic group:Z7
Dihedral group:D16	Cyclic group:Z4
Dihedral group:D18	Cyclic group:Z9
Dihedral group:D20	Cyclic group:Z5
Dihedral group:D22	Cyclic group:Z11
Dihedral group:D24	Cyclic group:Z6
Dihedral group:D26	Cyclic group:Z13
Dihedral group:D28	Cyclic group:Z7
Dihedral group:D30	Cyclic group:Z15

See also

• Derived subgroup of dihedral group:D16