# Lagrange's Theorem (Group Theory)/Proof 2

## Contents

## 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.

## Proof

Let $G$ be a group.

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

From Cosets are Equivalent, a left coset $y H$ has the same number of elements as $H$, namely $\order H$.

Since left cosets are identical or disjoint, each element of $G$ belongs to exactly one left coset.

From the definition of index of subgroup, there are $\index G H$ left cosets, and therefore:

- $\order G = \index G H \order H$

Let $G$ be of finite order.

All three numbers are finite, and the result follows.

Now let $G$ be of infinite order.

If $\index G H = n$ is finite, then $\order G = n \order H$ and so $H$ is of infinite order.

If $H$ is of finite order such that $\order H = n$, then $\order G = \index G H \times n$ and so $\index G H$ is infinite.

$\blacksquare$

## Source of Name

This entry was named for Joseph Louis Lagrange.

## Sources

- 1965: J.A. Green:
*Sets and Groups*... (previous) ... (next): $\S 6.4$. Lagrange's theorem - 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Subgroups and Cosets: $\S 40$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 43.1$ Lagrange's theorem - 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $0$: Some Conventions and some Basic Facts - 1982: P.M. Cohn:
*Algebra Volume 1*(2nd ed.) ... (previous) ... (next): $\S 3.3$: Group actions and coset decompositions: Theorem $1$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $5$: Cosets and Lagrange's Theorem: Theorem $5.9$