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.