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
Let us set up mappings $$\theta: H \to H x$$ and $$\phi: H x \to H$$ as follows:
 * $$\forall u \in H: \theta \left({u}\right) = u x$$
 * $$\forall v \in H x: \phi \left({v}\right) = v x^{-1}$$

Note that $$v \in H x \implies v x^{-1} \in H$$ from Elements in Coset iff Product with Inverse in Coset.

Now:
 * $$\forall v \in H x: \theta \circ \phi \left({v}\right) = v x^{-1} x = v$$
 * $$\forall u \in H: \phi \circ \theta \left({u}\right) = u x x^{-1} = u$$

Thus $$\theta \circ \phi = I_{Hx}$$ and $$\phi \circ \theta = I_H$$.

So $$\theta = \phi^{-1}$$: both are bijections and one is the inverse of the other.

Hence the result.

Alternatively, it follows directly from Set Equivalence of Regular Representations.