Coset Space forms Partition/Examples/Dihedral Group D3/Cosets of Subgroup Generated by b

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Examples of Use of Coset Space forms Partition

Consider the dihedral group $D_3$.

$D_3 = \gen {a, b: a^3 = b^2 = e, a b = b a^{-1} }$

From Dihedral Group $D_3$: Cosets of $\gen b$, the left cosets of of the subgroup $\gen b$ generated by $b$ are:

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

It follows from Coset Space forms Partition that these are consequences of:

\(\ds b^{-1} e\) \(=\) \(\ds b^{-1} = b \in H\)
\(\ds \paren {a b}^{-1} a = b^{-1} a^{-1} a\) \(=\) \(\ds b^{-1} = b \in H\)
\(\ds \paren{a^2 b}^{-1} a^2 = b^{-1} a^{-2} a^2\) \(=\) \(\ds b^{-1} = b \in H\)