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

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Examples of 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:

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


Left Cosets

The left cosets of $H$ are:

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


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


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


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


Right Cosets

The right cosets of $H$ are:

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


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


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


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


It is seen that the left cosets do not equal the corresponding right cosets.

It follows by definition that $\gen b$ is not a normal subgroup of $D_4$.


Sources

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