(related to Lemma: Subgroups and Their Cosets are Equipotent)

- Let $(G,\ast)$ be a group, $a\in G$ and let $H\subseteq G$ be a subgroup.
- For the left coset $aH$ let $f:H\to aH,$ $f(x)=ax$ be a function.
- By definition, $f$ is surjective.
- If $f(x)=f(y)$ then $ax=ay$ and since $a$ has an inverse in $G,$ also $x=y.$
- Thus, $f$ is injective.
- Thus, $H$ and $aH$ are equipotent.
- The same reasoning holds for right cosets.∎

**Modler, Florian; Kreh, Martin**: "Tutorium Algebra", Springer Spektrum, 2013