Coset/Examples/Dihedral Group D3/Cosets of Subgroup Generated by b/Left Cosets
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Examples of Left 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:
- $\gen b = \set {e, b}$
The left cosets of $H$ are:
\(\ds e H\) | \(=\) | \(\ds \set {e, b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds b H\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H\) |
\(\ds a H\) | \(=\) | \(\ds \set {a, a b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a b H\) |
\(\ds a^2 H\) | \(=\) | \(\ds \set {a^2, a^2 b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a^2 b H\) |
Proof
The Cayley table of $D_3$ is presented as:
- $\begin{array}{c|cccccc}
& e & a & a^2 & b & a b & a^2 b \\
\hline e & e & a & a^2 & b & a b & a^2 b \\ a & a & a^2 & e & a b & a^2 b & b \\ a^2 & a^2 & e & a & a^2 b & b & a b \\ b & b & a^2 b & a b & e & a^2 & a \\ a b & a b & b & a^2 b & a & e & a^2 \\ a^2 b & a^2 b & a b & b & a^2 & a & e \\ \end{array}$
Thus:
\(\ds e H\) | \(=\) | \(\ds e \set {e, b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {e^2, e b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {e, b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H\) |
\(\ds b H\) | \(=\) | \(\ds b \set {e, b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {b e, b^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {b, e}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds H\) |
\(\ds a H\) | \(=\) | \(\ds a \set {e, b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {a e, a b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {a, a b}\) |
\(\ds a^2 H\) | \(=\) | \(\ds a^2 \set {e, b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {a^2 e, a^2 b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {a^2, a^2 b}\) |
\(\ds a b H\) | \(=\) | \(\ds a b \set {e, b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {a b e, a b b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {a b, a}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a H\) |
\(\ds a^2 b H\) | \(=\) | \(\ds a^2 b \set {e, b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {a^2 b e, a^2 b b}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {a^2 b, a^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a^2 H\) |
$\blacksquare$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $5$: Cosets and Lagrange's Theorem: Example $5.2$