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

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