Left and Right Coset Spaces are Equivalent/Proof 1

Theorem
Let $\left({G, \circ}\right)$ be a group.

Let $H$ be a subgroup of $G$.

Let:
 * $x H$ denote the left coset of $H$ by $x$
 * $H y$ denote the right coset of $H$ by $y$.

Then:
 * $\left|{\left\{{x H: x \in G}\right\}}\right| = \left|{\left\{{H y: y \in G}\right\}}\right|$

To put it another way:
 * The number of right cosets is the same as the number of left cosets of $G$ with respect to $H$.


 * The left and right coset spaces are equivalent.

Proof
Let there be exactly $r$ different left cosets of $H$ in $G$.

Let a complete repetition-free list of these left cosets be:
 * $a_1 H, a_2 H, a_3 H \ldots, a_r H: a_1, a_2, \ldots, a_r \in G$

From Coset Spaces form Partition, every element of $G$ is contained in exactly one of the left cosets.

Let $x \in G$. Then, for $1 \le i \le r$:

Since $x^{-1} \in a_i H$ is true for precisely one value of $i$, it follows that $x \in H a_i^{-1}$ is also true for precisely that value of $i$.

So there are exactly $r$ different right cosets of $H$ in $G$, and a complete repetition-free list of these is:


 * $H a_1^{-1}, H a_2^{-1}, H a_3^{-1}, \ldots, H a_r^{-1}$

The result follows.