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:
\(\ds x \equiv^l y \pmod H\) | \(\iff\) | \(\ds x H = y H\) | ||||||||||||
\(\ds \neg \paren {x \equiv^l y} \pmod H\) | \(\iff\) | \(\ds x H \cap y H = \O\) |
Right Coset Space forms Partition
The right coset space of $H$ forms a partition of its group $G$:
\(\ds x \equiv^r y \pmod H\) | \(\iff\) | \(\ds H x = H y\) | ||||||||||||
\(\ds \neg \paren {x \equiv^r y} \pmod H\) | \(\iff\) | \(\ds H x \cap H y = \O\) |
Examples
Dihedral Group $D_3$: Cosets of $\gen b$
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:
\(\ds b^{-1} e\) | \(=\) | \(\ds b^{-1} = b \in H\) | ||||||||||||
\(\ds \paren {a b}^{-1} a = b^{-1} a^{-1} a\) | \(=\) | \(\ds b^{-1} = b \in H\) | ||||||||||||
\(\ds \paren{a^2 b}^{-1} a^2 = b^{-1} a^{-2} a^2\) | \(=\) | \(\ds b^{-1} = b \in H\) |
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): coset