Coset Space forms Partition

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Theorem

Let $G$ be a group, and let $H \le G$.

Left Coset Space forms Partition

The left coset space of $H$ forms a partition of its group $G$, and hence:

\(\displaystyle x \equiv^l y \pmod H\) \(\iff\) \(\displaystyle x H = y H\)
\(\displaystyle \neg \paren {x \equiv^l y} \pmod H\) \(\iff\) \(\displaystyle x H \cap y H = \O\)


Right Coset Space forms Partition

The right coset space of $H$ forms a partition of its group $G$:

\(\displaystyle x \equiv^r y \pmod H\) \(\iff\) \(\displaystyle H x = H y\)
\(\displaystyle \neg \paren {x \equiv^r y} \pmod H\) \(\iff\) \(\displaystyle H x \cap H y = \O\)


Examples

Dihedral Group $D_3$: Cosets of $\gen b$

<onlyinclude> 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:

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


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

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