Order of Groups

The order of a group is the number of its elements; the order of an element is the number of times you have to apply it to get back to the identity.

Definition

Let G be a group.

• The order of G, written |G|, is the cardinality of the underlying set. G is finite when |G| is finite. This site focuses on finite groups.
• The order of an element a ∈ G, written |a| or ord(a), is the smallest positive integer n with aⁿ = e. If no such n exists, a has infinite order. In a finite group every element has finite order.

The exponent of G is the least common multiple of the orders of its elements — equivalently, the smallest n with aⁿ = e for all a ∈ G.

Examples

• In C₁₂ (rotations by multiples of 30°), the rotation by 30° has order 12, by 60° has order 6, by 180° has order 2.
• |Sₙ| = n!. A transposition in S₄ has order 2; an n-cycle has order n.
• |Dₙ| = 2n. In D₄, the rotation generator has order n; every reflection has order 2.
• |GL(2,𝔽ₚ)| = (p² − 1)(p² − p).

Properties

• An element's order divides the group order. |a| divides |G| — apply Lagrange's theorem to the cyclic subgroup ⟨a⟩.
• An element's order is the size of the cyclic subgroup it generates: |⟨a⟩| = |a|, since ⟨a⟩ = {e, a, a², …, a|a|−1}.
• A finite group's exponent divides its order. exp(G) | |G|, with equality exactly when G has an element of order |G| — that is, when G is cyclic.
• The order of a power: |aᵏ| = |a| / gcd(|a|, k) for any element a of finite order.
• Every element raised to the group order is the identity: a|G| = e for every a in a finite group G.
• Every group of prime order is cyclic. If |G| = p is prime, a non-identity element has order dividing p, hence order p, so it generates G.
• Lagrange's theorem: in a finite group, |H| divides |G| for every subgroup H ≤ G.

Computation

• Polycyclic group. |G| is the product of relative orders of the pc generators — read off in O(n) time where n is the number of generators.
• Permutation group. Compute a base and strong generating set (BSGS) via Schreier–Sims; then |G| is the product of orbit sizes along the base.
• Order of an element. In a cyclic or pc setting, compute from the generator representation. In a permutation group, |a| is the least common multiple of its cycle lengths.

See also

• Subgroup — where ⟨a⟩ lives.
• Lagrange's theorem — why element order divides group order.
• Abelian group — when exponent equals the largest elementary divisor.