Cyclic and Finitely Generated Abelian Groups

Abelian groups are the part of group theory that is completely solved: every finitely generated abelian group is a direct sum of cyclic groups, and the decomposition is unique. This page classifies cyclic groups and states the fundamental theorem of finitely generated abelian groups, the abelian bedrock under Sylow theory and the weight lattices of reps-lie.md. It builds on axioms.md and products.md.

Cyclic groups

A group is cyclic if generated by a single element, $G=⟨g⟩$. There are exactly two isomorphism types:

• Infinite cyclic $\cong \mathbb{Z}$ (if $\mathrm{ord}(g)=\infty$).
• Finite cyclic $\cong {\mathbb{Z}}_n$ (if $\mathrm{ord}(g)=n$): the integers mod $n$ under addition, equivalently the $n$-th roots of unity.

Structure of ${\mathbb{Z}}_n$.

• Every subgroup is cyclic; there is exactly one subgroup of each order $d\mid n$, namely $⟨n/d⟩\cong {\mathbb{Z}}_d$. Subgroups ↔ divisors.
• The generators of ${\mathbb{Z}}_n$ are the residues coprime to $n$; there are $\phi (n)$ of them (Euler's totient).
• ${\mathbb{Z}}_m\times {\mathbb{Z}}_n\cong {\mathbb{Z}}_{mn}$ iff $\gcd (m,n)=1$ — the Chinese remainder theorem in group form. This is the tool that breaks cyclic groups into prime-power pieces.

Cyclic groups are the abelian atoms and the simplest Lie-group analogue is $U(1)$, whose character theory (Fourier series) is the continuous mirror of the finite characters below.

The fundamental theorem

Fundamental theorem of finitely generated abelian groups. Every finitely generated abelian group $A$ is a finite direct sum $$A\cong {\mathbb{Z}}^r\oplus {\mathbb{Z}}_{d_1}\oplus \cdots \oplus >{\mathbb{Z}}_{d_k},$$ and the decomposition can be taken in two canonical forms:

• Invariant-factor form: $d_1\mid d_2\mid \cdots \mid d_k$ (each divides the next); the $d_i$ and the rank $r$ are uniquely determined by $A$.
• Primary form: a direct sum of cyclic groups of prime-power order ${\mathbb{Z}}_{p^a}$; the multiset of prime powers is uniquely determined.

The two forms are interconvertible by the Chinese remainder theorem. The integer $r$ is the free rank; the finite part ${\mathbb{Z}}_{d_1}\oplus \cdots$ is the torsion subgroup (elements of finite order).

Consequence — counting abelian groups. The number of abelian groups of order $n=\prod p_i^{a_i}$ is ${\prod }_{i}P(a_i)$, where $P$ is the integer-partition function: the abelian groups of order $p^a$ correspond to the partitions of $a$ (each partition $a=\sum b_j$ giving ${⨁}_j{\mathbb{Z}}_{p^{b_j}}$). For example the abelian groups of order $8=2^3$ are ${\mathbb{Z}}_8$, ${\mathbb{Z}}_4\oplus {\mathbb{Z}}_2$, ${\mathbb{Z}}_2^3$ — the three partitions of $3$.

Characters and Pontryagin duality

The irreducible representations of a finite abelian group $A$ are all $1$-dimensional (Schur's lemma — see reps-basics.md); they are the characters $\chi :A\to U(1)$. The characters themselves form a group $\hat{A}$ under pointwise multiplication, the Pontryagin dual, with $$\hat{A}\cong A(\text{non-canonically}),\widehat{{\mathbb{Z}}_n}\cong {\mathbb{Z}}_n,\hat{\mathbb{Z}}\cong U(1).$$ Expanding functions on $A$ in characters is the finite Fourier transform; the continuous version for $U(1)$ is ordinary Fourier series, and both are instances of the Peter–Weyl theorem. Character theory for non-abelian groups — where irreps are higher-dimensional — is character-theory.md.

References

• Dummit & Foote, Abstract Algebra, §5.2.
• Artin, Algebra, Ch. 14 (abelian groups and modules).