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