Lagrange's Theorem

Introduction

This partitioning of a finite group into equal-sized pieces leads directly to a powerful and elegant result about the orders of subgroups. Lagrange's Theorem is a cornerstone of finite group theory.

Statement

Theorem 2.2 (Lagrange's Theorem): If $G$ is a finite group and $H$ is a subgroup of $G$ , then the order of $H$ divides the order of $G$ .

Proof

Let $| G | = m$ and $| H | = n$ . The distinct left cosets of $H$ in $G$ partition the set $G$ . Let there be $p$ distinct left cosets. Since each coset has the same number of elements as $H$ , namely $n$ , the total number of elements in $G$ is the sum of the elements in all the distinct cosets. Therefore, $m = n p$ . This shows that $n$ divides $m$ .

Index of a Subgroup

The number of distinct cosets, $p$ , is called the index of $H$ in $G$ , denoted $[G : H]$ . The theorem can thus be stated as:

| G | = [G : H] \cdot | H |

Consequences

Lagrange's Theorem is a cornerstone of finite group theory because it places a strong restriction on the possible sizes of subgroups. For instance, a group of order 30 can only have subgroups of orders 1, 2, 3, 5, 6, 10, 15, and 30.

Important Note: Converse is False

The theorem is a one-way implication. The converse is not true in general: if $d$ is a divisor of $| G |$ , there is not necessarily a subgroup of order $d$ .

Counterexample: The alternating group $A_{4}$ has order 12 but contains no subgroup of order 6. This failure of the converse of Lagrange's Theorem is a deep and important feature of group structure, and it serves as a primary motivation for the Sylow Theorems, which provide a partial converse for orders that are powers of a prime.

Examples

Example 1: Groups of Order 6

A group of order 6 can only have subgroups of orders 1, 2, 3, and 6. The possible subgroups are:

${e}$ (order 1)
Any subgroup of order 2 (if it exists)
Any subgroup of order 3 (if it exists)
The group itself (order 6)

Example 2: Groups of Order 8

A group of order 8 can only have subgroups of orders 1, 2, 4, and 8. Examples include:

$Z_{8}$ : has subgroups of orders 1, 2, 4, 8
$D_{4}$ : has subgroups of orders 1, 2, 4, 8
$Q_{8}$ (quaternion group): has subgroups of orders 1, 2, 4, 8

Example 3: Groups of Order 12

A group of order 12 can only have subgroups of orders 1, 2, 3, 4, 6, and 12. However, not all of these may exist:

$Z_{12}$ : has subgroups of all possible orders
$A_{4}$ : has subgroups of orders 1, 2, 3, 4, 12 (but not 6)

Applications

Application 1: Order of Elements

Lagrange's Theorem can be used to find the order of elements. If $g \in G$ has order $k$ , then $⟨ g ⟩$ is a subgroup of order $k$ , so $k$ must divide $| G |$ .

Example: In a group of order 15, any element must have order 1, 3, 5, or 15.

Application 2: Prime Order Groups

If $G$ is a group of prime order $p$ , then by Lagrange's Theorem, the only possible subgroups are ${e}$ and $G$ itself. This implies that $G$ is cyclic and generated by any non-identity element.

Example: Any group of order 7 is cyclic and isomorphic to $Z_{7}$ .

Application 3: Index Calculations

The index $[G : H]$ can be calculated as $[G : H] = | G | / | H |$ , which is often useful in counting arguments.

Example: If $G$ has order 24 and $H$ has order 6, then $[G : H] = 24 / 6 = 4$ .

Application 4: Existence of Elements

Lagrange's Theorem can be used to prove the existence of elements with certain properties.

Example: In a group of order $p^{2}$ where $p$ is prime, there must be an element of order $p$ (since the group cannot be cyclic of order $p^{2}$ ).

Cauchy's Theorem

For a finite group $G$ and a prime $p$ dividing $| G |$ , there exists an element of order $p$ in $G$ .

Sylow Theorems

These provide a partial converse to Lagrange's Theorem for prime power orders.

Summary

Lagrange's Theorem provides a fundamental restriction on the structure of finite groups by limiting the possible sizes of subgroups. While the converse is not true, this theorem is essential for understanding group structure and serves as a foundation for more advanced results in group theory.