# 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.