Cyclic groups

A group is cyclic when a single element generates it: G = ⟨g⟩. Up to isomorphism there is exactly one cyclic group of each order n, written Cₙ.

Definition

For n ≥ 1,

Cₙ = ⟨g | gⁿ = 1⟩ = {1, g, g², …, gⁿ⁻¹},    |Cₙ| = n.

Cₙ ≅ ℤ/nℤ, the integers mod n under addition (g ↔ 1). A group generated by one element of infinite order is the infinite cyclic group ℤ.

Structure

Abelian. gᵃ gᵇ = g^{a+b} = gᵇ gᵃ, so every cyclic group is abelian.
Subgroups. Every subgroup of a cyclic group is cyclic. Cₙ has exactly one subgroup of each order d | n — namely ⟨g^n/d⟩, of order d — and no others. Hence the subgroup lattice of Cₙ is the divisor lattice of n, ordered by divisibility.
Prime factorization. If n = p₁^a₁ ⋯ p_k^a_k, the Chinese remainder theorem gives
```
Cₙ ≅ C_p₁^a₁ × ⋯ × C_{p_k^a_k},
```
the direct product of the Sylow subgroups of Cₙ — so Cₙ is nilpotent.

Generators and element orders

gᵏ generates Cₙ iff gcd(k, n) = 1, so Cₙ has exactly φ(n) generators (Euler's totient).
For each d | n, Cₙ has exactly φ(d) elements of order d, all lying in the unique order-d subgroup; summing over divisors gives ∑_d|n φ(d) = n.
Automorphisms. Aut(Cₙ) ≅ (ℤ/nℤ)ˣ, the multiplicative group of units mod n, of order φ(n): an automorphism sends g ↦ gᵏ with gcd(k, n) = 1.

Simplicity and characterization

Cₚ for p prime is simple; the Cₚ are the only abelian simple groups.
Every group of prime order p is cyclic (≅ Cₚ): a non-identity element has order dividing p, hence order p, hence generates.
A finite group G is cyclic iff exp(G) = |G| — iff it has an element of order |G|.

Properties

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.
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.
Every cyclic group is abelian.
A cyclic group splits into prime-power cyclic factors: if n = pᵃ qᵇ ⋯ then C_n ≅ C_pᵃ × C_qᵇ × ⋯. Every finite nilpotent group similarly splits into its Sylow (prime-power) subgroups; for abelian groups those factors break further into cyclic ones.
A cyclic group of prime order is simple: its only subgroups are {e} and itself, and these Cₚ are the only abelian simple groups.

Examples

C₁ (trivial), C₂, C₃ — the smallest cyclic groups.
Order 4. The two groups of order 4 are C₄ and the Klein four-group C₂ × C₂; only C₄ is cyclic (C₂ × C₂ has exponent 2 < 4, so no single generator).
C₆ ≅ C₂ × C₃ — the smallest nontrivial CRT split.
(ℤ/5ℤ)ˣ ≅ C₄ — the units mod 5, generated by 2 (2, 4, 3, 1).