Dihedral groups are non-abelian

https://arbital.com/p/dihedral_groups_are_non_abelian

by Patrick Stevens Jun 15 2016 updated Jun 15 2016

The group of symmetries of the triangle and all larger regular polyhedra are not abelian.


Let . Then the Dihedral group on vertices, , is not abelian.

Proof

The most natural dihedral group presentation is . In particular, , so if and only if is the identity. But is the rotation which has order , so cannot be equal to .

[todo: picture with the triangle]