Cosets are Equivalent

Theorem
All left cosets of a group $$G$$ with respect to a subgroup $$H$$ are equivalent.

That is, any two left cosets are in one-to-one correspondence.

The same applies to right cosets.

As a special case of this:
 * $$\forall x \in G: \left|{x H}\right| = \left|{H}\right| = \left|{H x}\right|$$

where $$H$$ is a subgroup of $$G$$.

Proof
This follows directly from Set Equivalence of Regular Representations.