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

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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:

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

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

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