Cosets have the same cardinality as the subgroup

Statement

Let $G$ be a group, let $H$ be a subgroup of $G$, and let $g \in G$. Then the map $h \mapsto gh$ is a bijection from $H$ to the left coset $gH$. The map $h \mapsto hg$ is a bijection from $H$ to the right coset $Hg$. In particular, $|gH| = |Hg| = |H|$.

Remarks

The inverse of $h \mapsto gh$ is $x \mapsto g^{-1}x$. This fact and the partition of $G$ into left cosets give the proof of Lagrange's theorem.

Explanations

L1 human

Multiplying by a fixed element never makes two different elements equal, by cancellation. Thus a coset has exactly as many elements as the subgroup. All cosets of a subgroup have the same size.