Derived subgroup of dihedral group

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

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

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

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

Table of examples

See also