Lagrange's Theorem

Theorem: Lagrange's Theorem

Let $G$ be a finite group. Let $H\subseteq G$ be a subgroup of $G$. Then $\left|H\right|$ divides $\left|G\right|$.

Proof:

We can prove this using Left Cosets

Because all left cosets of $H$ have the same order, and $G$ is the disjoint union of all left cosets of $H$ as below: $G={⨆}_{a\in G}aH,$ we can say that $\left|G\right|=\left|H\right|\cdot \left(\text{number of disjoint left cosets}\right),$, which is also denoted $\left|G\right|=\left|H\right|\cdot \left[G:H\right],$

Corollary

Let $a\in G$. Then $o(a)$ divides $\left|G\right|$.