Dihedral Group D4/Subgroups/Cosets/Subgroup Generated by b/Right Cosets

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Examples of Left Cosets of Subgroups of Dihedral Group $D_4$

Let the dihedral group $D_4$ be represented by its group presentation:

$D_4 = \gen {a, b: a^4 = b^2 = e, a b = b a^{-1} }$


Let $H \subseteq D_4$ be defined as:

$H = \gen b$

where $\gen b$ denotes the subgroup generated by $b$.

From Subgroups of Dihedral Group D4 we have:

$\gen b = \set {e, b}$


The right cosets of $H$ are:

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


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


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


\(\ds H a^3\) \(=\) \(\ds \set {a^3, b a^3}\)
\(\ds \) \(=\) \(\ds H b a^3\)


Proof

The Cayley table of $D_4$ is presented as:

$\begin{array}{l|cccccccc}
      &     e &     a &   a^2 &   a^3 &     b &   b a & b a^2 & b a^3 \\

\hline 

e     &     e &     a &   a^2 &   a^3 &     b &   b a & b a^2 & b a^3 \\
a     &     a &   a^2 &   a^3 &     e & b a^3 &     b &   b a & b a^2 \\
a^2   &   a^2 &   a^3 &     e &     a & b a^2 & b a^3 &     b &   b a \\
a^3   &   a^3 &     e &     a &   a^2 &   b a & b a^2 & b a^3 &     b \\
b     &     b &   b a & b a^2 & b a^3 &     e &     a &   a^2 &   a^3 \\
b a   &   b a & b a^2 & b a^3 &     b &   a^3 &     e &     a &   a^2 \\
b a^2 & b a^2 & b a^3 &     b &   b a &   a^2 &   a^3 &     e &     a \\
b a^3 & b a^3 &     b &   b a & b a^2 &     a &   a^2 &   a^3 &     e

\end{array}$


Thus:

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


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


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


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


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


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


\(\ds H a^3\) \(=\) \(\ds \set {e, b} a^3\)
\(\ds \) \(=\) \(\ds \set {e a^3, b a^3}\)
\(\ds \) \(=\) \(\ds \set {a^3, b a^3}\)


\(\ds H b a^3\) \(=\) \(\ds \set {e, b} b a^3\)
\(\ds \) \(=\) \(\ds \set {e b a^3, b b a^3}\)
\(\ds \) \(=\) \(\ds \set {b a^3, a^3}\)
\(\ds \) \(=\) \(\ds H a^3\)

$\blacksquare$


Sources

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