Coset/Examples/Dihedral Group D3/Cosets of Subgroup Generated by b

Examples of Cosets
Consider the dihedral group $D_3$.

Let $H \subseteq D_3$ be defined as:
 * $H = \gen b$

where $\gen b$ denotes the subgroup generated by $b$.

As $b$ has order $2$, it follows that:
 * $\gen b = \set {e, b}$

The left cosets of $H$ are:

Proof
The Cayley table of $D_3$ is presented as:

Thus: