Dihedral Group D4/Subgroups/Cosets
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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:
- $D_4 = \gen {a, b: a^4 = b^2 = e, a b = b a^{-1} }$
Generated Subgroup $\gen b$
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 $D_4$.