A4 - symmetries of a square

a square can be transformed in 8 ways such that it is unchanged:
- 4 counter-clockwise rotations: $R_{0}, R_{90}, R_{180}, R_{270}$
- 4 reflections: $H, V, D, D^{'}$
these form the dihedral group $D_{4}$ under

the identity is $R_{0}$ , and the group is non-abelian

dihedral group

$D_{n}$ is the group of symmetries for a regular $n$ -gon, which includes $n$ rotations and $n$ reflections

it is finite and non-abelian

![info] cyclic rotation group

$R_{n}$ is a symmetry group comprising of rotations of an $n$ -gon with members:

$⟨ R_{360 / n} ⟩ := 0, \frac{360}{n}, 2 \frac{360}{n}, \dots (n - 1) \frac{360}{n}$