Coset Space forms Partition

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