# Lagrange theorem on subgroup size

Lagrange’s Theorem states that if \(G\) is a finite group and \(H\) a subgroup, then the order \(|H|\) of \(H\) divides the order \(|G|\) of \(G\). It generalises to infinite groups: the statement then becomes that the left cosets form a partition, and for any pair of cosets, there is a bijection between them.

# Proof

In full generality, the cosets form a partition and are all in bijection.

To specialise this to the finite case, we have divided the \(|G|\) elements of \(G\) into buckets of size \(|H|\) (namely, the cosets), so \(|G|/|H|\) must in particular be an integer.

Children:

- Lagrange theorem on subgroup size: Intuitive version
Lagrange’s theorem strongly restricts the size a subgroup of a group can be.

Parents:

- Group
The algebraic structure that captures symmetry, relationships between transformations, and part of what multiplication and addition have in common.