# Dihedral groups are non-abelian

Let $$n \geq 3$$. Then the dihedral group on $$n$$ vertices, $$D_{2n}$$, is not abelian.

# Proof

The most natural dihedral group presentation is $$\langle a, b \mid a^n, b^2, bab^{-1} = a^{-1} \rangle$$. In particular, $$ba = a^{-1} b = a^{-2} a b$$, so $$ab = ba$$ if and only if $$a^2$$ is the identity. But $$a$$ is the rotation which has order $$n > 2$$, so $$ab$$ cannot be equal to $$ba$$.

picture with the triangle

Parents:

• Dihedral group

The dihedral groups are natural examples of groups, arising from the symmetries of regular polygons.