Abelian groups

A group is abelian when its operation commutes: ab = ba for all a, b. Every finite abelian group is a direct product of cyclic groups of prime-power order, unique up to ordering.

Definition

A group G is abelian (or commutative) if

ab = ba   for all a, b ∈ G.

Equivalently, [G, G] = {e}, or Z(G) = G (see commutator, center).

Fundamental theorem of finite abelian groups

Every finite abelian group is isomorphic to a direct product of cyclic groups of prime-power order. This decomposition is unique up to reordering. The decomposition admits two standard normal forms:

Elementary divisor form.

G  ≅  C_p₁^a₁ × C_p₂^a₂ × ⋯ × C_pₖ^aₖ,

where the pᵢ are primes (not necessarily distinct) and aᵢ ≥ 1. This is the form we use as the canonical page URL: /a/8_4_3 denotes C₈ × C₄ × C₃.

Invariant factor form.

G  ≅  C_d₁ × C_d₂ × ⋯ × C_dₘ,     with  d₁ | d₂ | ⋯ | dₘ.

dₘ is the exponent of G; d₁ is the smallest invariant factor.

The two forms are interconvertible: group the elementary divisors by prime, align the prime-power sequences by length, and multiply across primes to recover the invariant factors.

Subgroups

The subgroups of an abelian group of prime-power order depend on the exponents alone, not on the prime. For many shapes the whole subgroup lattice is therefore one object valid at every prime: the number of subgroups of each kind is a polynomial in p, and substituting a prime gives the lattice of a concrete group.

Subgroup lattices of abelian p-groups — every shape with such a description, its classes of subgroups, and its diagram.
C₈ × C₄ × C₂ and C₂₇ × C₉ × C₃ are the same shape read at p = 2 and p = 3.

Examples

C₆ ≅ C₂ × C₃ — the simplest case of the Chinese remainder theorem.
Order 4. Exactly two abelian groups: C₄ and the Klein four-group V = C₂ × C₂. See all abelian groups of order 4.
Order 8. Three abelian groups: C₈, C₄ × C₂, C₂ × C₂ × C₂. See all abelian groups of order 8.
(ℤ/nℤ)ˣ — the unit group mod n. Abelian. Explicit decomposition via CRT and the structure of (ℤ/pᵏℤ)ˣ.

Counting

The number of abelian groups of order n is

a(n)  =  ∏_p  P(a_p),

where n = ∏ p^a_p and P(k) is the number of integer partitions of k. Thus a(pⁿ) = P(n): a(2) = 1, a(4) = 2, a(8) = 3, a(16) = 5, a(32) = 7, a(64) = 11.

Properties

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.
Every subgroup and every quotient of an abelian group is again abelian.
Every irreducible complex representation of a finite abelian group is 1-dimensional; its character table is a (generalized) discrete Fourier transform matrix — the ordinary DFT matrix when the group is cyclic.
A finite abelian group's subgroup lattice is self-dual: swapping every subgroup with a matching quotient and reversing the cover relation reproduces the same diagram.
A finite nilpotent group's subgroup lattice is the direct product of its Sylow subgroups' lattices — abelian groups included.

Computation

From generators and relations. Given a finitely presented abelian group G = ⟨x₁, …, xₙ | R⟩ (relations in additive form), form the relation matrix M and compute its Smith normal form to read off the invariant factors.
Abelianization of a general group. Compute G/[G, G] — the result is abelian and can be normalized as above; see commutator.
Direct product decomposition. Group the elementary divisors by prime; each p-part is a direct sum of cyclic p-groups.