Lagrange's theorem

The order of a subgroup divides the order of the group. This is the first non-trivial structural theorem in group theory and the reason why order arithmetic constrains what groups can exist.

Definition

Let H ≤ G be a subgroup of a finite group.

The left coset of H by g is gH = {gh : h ∈ H}.
The right coset is Hg = {hg : h ∈ H}.
The index [G : H] is the number of distinct left cosets of H in G (equal to the number of right cosets).

The left cosets partition G: distinct cosets are disjoint, and every element lies in exactly one coset. Each coset has exactly |H| elements (the map h ↦ gh is a bijection H → gH).

Theorem

Lagrange's theorem. For any subgroup H of a finite group G,

|G| = |H| · [G : H].

In particular, |H| divides |G|.

Proof sketch. The left cosets partition G into [G : H] blocks of size |H|; count. ∎

Corollaries

Element order divides group order. For any a ∈ G, |a| divides |G| — apply Lagrange to the cyclic subgroup ⟨a⟩.
a^|G| = e for every a ∈ G.
Groups of prime order are cyclic. If |G| = p prime, then any non-identity element has order p (by 1), so it generates G.

The converse fails

Lagrange tells us which subgroup orders are possible, not which actually occur. A divisor of |G| need not be the order of any subgroup.

A₄ has order 12 but no subgroup of order 6. If it had one, that subgroup would have index 2, hence be normal, but A₄ has no normal subgroup of order 6.
Sylow's theorems partially recover a converse for prime-power divisors: a subgroup of order pᵏ exists whenever pᵏ divides |G|.

Computation