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.

Also see

 * Group does not Necessarily have Subgroup of Order of Divisor of its Order, where it is shown that the converse is not true in general.