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

Examples of Cosets

Consider the dihedral group $D_3$.

$D_3 = \gen {a, b: a^3 = b^2 = e, a b = b a^{-1} }$

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:

$\gen b = \set {e, b}$

Left Cosets

The left cosets of $H$ are:

\(\ds e H\)

\(=\)

\(\ds \set {e, b}\)

\(\ds \)

\(=\)

\(\ds b H\)

\(\ds \)

\(=\)

\(\ds H\)

\(\ds a H\)

\(=\)

\(\ds \set {a, a b}\)

\(\ds \)

\(=\)

\(\ds a b H\)

\(\ds a^2 H\)

\(=\)

\(\ds \set {a^2, a^2 b}\)

\(\ds \)

\(=\)

\(\ds a^2 b H\)

Right Cosets

The right cosets of $H$ are:

\(\ds H e\)

\(=\)

\(\ds \set {e, b}\)

\(\ds \)

\(=\)

\(\ds H b\)

\(\ds \)

\(=\)

\(\ds H\)

\(\ds H a\)

\(=\)

\(\ds \set {a, a^2 b}\)

\(\ds \)

\(=\)

\(\ds H a^2 b\)

\(\ds H a^2\)

\(=\)

\(\ds \set {a^2, a b}\)

\(\ds \)

\(=\)

\(\ds H a b\)