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Proof

(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.
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References

Bibliography

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