Lagrange's Theorem (Group Theory)/Proof 1

Proof
Let $G$ be finite.

Consider the mapping $\phi: G \to G / H^l$, defined as:


 * $\phi: G \to G / H^l: \phi \left({x}\right) = x H^l$

where $G / H^l$ is the left coset space of $G$ modulo $H$.

For every $y H \in G / H^l$, there exists a corresponding $y \in G$, so $\phi$ is a surjection.

From Cardinality of Surjection it follows that $G / H^l$ is finite.

From Cosets are Equivalent, $G / H^l$ has the same number of elements as $H$.

We have that the $G / H^l$ is a partition of $G$.

It follows from Number of Elements in Partition that $\left[{G : H}\right] = \dfrac {\left|{G}\right|} {\left|{H}\right|}$