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

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Examples of Right 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 right cosets of $H$ are:

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


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


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


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:

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


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


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


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


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


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

$\blacksquare$


Sources

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