Group theory (algebra)/Lagrange's theorem/Fact/Proof

Proof

We consider the left cosets ${}gH:={\left\{gh\mid h\in H\right\}}$ for all ${}g\in G$. The mapping

$H\longrightarrow gH,h\longmapsto gh,$

is a bijection between ${}H$ and ${}gH$, so that all cosets have the same number of elements (namely ${}{\#\left(H\right)}$). The cosets form (as they are the equivalence classes) a partition of ${}G$. Hence, ${}{\#\left(G\right)}$ is a multiple of ${}{\#\left(H\right)}$.

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