Lagrange's Theorem (Group Theory)

Theorem
Let $G$ be a finite group.

Let $H$ be a subgroup of $G$.

Then:
 * $\order H$ divides $\order G$

where $\order G$ and $\order H$ are the order of $G$ and $H$ respectively.

In fact:
 * $\index G H = \dfrac {\order G} {\order H}$

where $\index G H$ is the index of $H$ in $G$.

When $G$ is an infinite group, we can still interpret this theorem sensibly:


 * A subgroup of finite index in an infinite group is itself an infinite group.


 * A finite subgroup of an infinite group has infinite index.

Remark
The converse of Lagrange's theorem is not true in general.

We consider the symmetric group $S_4$.

Then the order of the alternating group $A_4$ is $12$.

Now $6$ divides $12$.

But there is no subgroup of $A_4$ of order $6$.