List of Abelian Groups

Introduction

This article lists the abelian groups the corpus meets, grouped by the structure theorem that classifies each family: the finitely generated groups, classified by a free rank and a torsion part; the torsion and the torsion-free groups, split by the torsion subgroup and the rank; the divisible groups, classified by their rank; and the profinite abelian groups, which carry a topology as well as a group law. Beside each family the list names the theorem that classifies it.

Every entry points to the article that introduces the group and names the invariant recorded there. This article is a list: it introduces no definition, states no theorem, gives no proof, and carries no display mathematics. It records examples and non-examples side by side, a non-example being an abelian group that fails one property of its family, with the failure named and the article that records it.

Finitely Generated Abelian Groups

A finitely generated abelian group is a direct sum of a free abelian group of finite rank and a finite abelian group, and the finite part is a direct sum of cyclic groups. The invariants are unique in two normal forms, the invariant factors $d_1 \mid \cdots \mid d_k$ and the elementary divisors $p_i^{e_i}$, and the theorem is the case $R = \mathbb{Z}$ of the structure theorem for finitely generated modules over a principal ideal domain.

Group	The property it has	Introduced in
$\mathbb{Z}^r$	free abelian of rank $r$; the free part of a finitely generated group	Finitely Generated Abelian Groups
$\mathbb{Z}/n\mathbb{Z}$	cyclic of order $n$, with $n$ decomposed into prime powers	Finitely Generated Abelian Groups
Finite abelian group	a direct sum of cyclic groups of prime-power order, uniquely	Finitely Generated Abelian Groups
$\mathbb{Z}/p^{e_1}\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/p^{e_k}\mathbb{Z}$	the elementary-divisor form of the $p$-primary part	Finitely Generated Abelian Groups
Group with invariant factors $d_1 \mid \cdots \mid d_k$	the divisibility-ordered form of the torsion part	Finitely Generated Abelian Groups
Relations matrix and Smith normal form	the algorithm computing the invariants from a presentation	Finitely Generated Abelian Groups
$(\mathbb{Z},+)$	free abelian of rank $1$; its torsion subgroup is trivial	Infinite Abelian Groups
Non-example: $\mathbb{Z}^r$ with $r \geq 1$	fails to be torsion: it has no nonzero element of finite order	Infinite Abelian Groups
Non-example: $\mathbb{Z}/n\mathbb{Z}$ with $n \geq 2$	fails to be torsion-free: every element has finite order	Infinite Abelian Groups
Non-example: the free abelian group of rank $\geq 2$	fails to be a free group: it satisfies $[a,b]=1$	Generators, Presentations and Free Products

Torsion and Torsion-Free Groups

The torsion subgroup $T(A)$ collects the elements of finite order, and $A$ is torsion, torsion-free or mixed according to what $T(A)$ is. The torsion groups decompose into their primary components $A_p$, and the countable primary groups are classified by the Ulm invariants; the torsion-free groups are classified by their rank only in the free case, the countable torsion-free groups admitting no comparably simple description.

Group	The property it has	Introduced in
Torsion subgroup $T(A)$	the elements of finite order; a subgroup	Infinite Abelian Groups
$A[p]$	the elements of order dividing $p$	Infinite Abelian Groups
Primary component $A_p$	the $p$-primary summand of a torsion group	Infinite Abelian Groups
Torsion group	a group with $T(A)=A$; a direct sum of its primary components	Infinite Abelian Groups
Torsion-free group	a group with $T(A)=0$; contains a free subgroup of the same rank	Infinite Abelian Groups
Rank $\operatorname{rk}(A)$	the cardinality of a maximal linearly independent set	Infinite Abelian Groups
Prüfer $p$-group $\mathbb{Z}[p^\infty]$	the $p$-primary torsion group of the $p^n$-th roots of unity	Infinite Abelian Groups
Ulm invariants $U(k,G)$	the complete invariants of a countable primary group	Infinite Abelian Groups
Non-example: a countable torsion-free group of rank $\geq 2$	fails to be classified by rank alone: the torsion-free classification is intractable	Infinite Abelian Groups
Non-example: $\mathbb{Q}$	fails to be free: it is torsion-free but has no basis over $\mathbb{Z}$	Infinite Abelian Groups

Divisible Groups

A divisible group is one in which every element is divisible by every positive integer; the divisible groups are classified by their rank as direct sums of copies of $\mathbb{Q}$ and of the Prüfer groups $\mathbb{Z}[p^\infty]$. Every group embeds in a divisible group, its divisible hull, which is unique up to isomorphism.

Group	The property it has	Introduced in
Divisible group $D$	every element is $n$-divisible for every $n$	Infinite Abelian Groups
$\mathbb{Q}$	the divisible torsion-free group of rank $1$; the divisible hull of $\mathbb{Z}$	Infinite Abelian Groups
$\mathbb{Q}/\mathbb{Z}$	the torsion divisible group, the direct sum of the Prüfer groups	Modules over a PID
$\mathbb{Z}[p^\infty]$	the Prüfer $p$-group; divisible and torsion	Infinite Abelian Groups
$D^{(I)}$	the direct sum of $\|I\|$ copies of the fixed rank-one divisible group $D$; the general divisible torsion-free group	Infinite Abelian Groups
Divisible hull	the smallest divisible group containing a given group; unique	Infinite Abelian Groups
Non-example: $\mathbb{Z}$	fails divisibility: $1$ is not divisible by $2$	Infinite Abelian Groups
Non-example: a finite abelian group	fails divisibility: no nonzero element is infinitely divisible	Infinite Abelian Groups

Profinite and Topological Abelian Groups

A profinite abelian group is an inverse limit of finite abelian groups, equivalently a compact totally disconnected abelian group; the group carries a topology as well as a group law, and the topological theory, including Pontryagin duality, belongs to Part II. The additive groups $\mathbb{Z}_p$ and the profinite completion $\hat{\mathbb{Z}}$ are the standard examples, and the Galois groups of the arithmetic articles are their non-abelian relatives.

Group	The property it has	Introduced in
$\mathbb{Z}_p$	the additive group of $p$-adic integers; profinite, torsion-free and not divisible	Profinite Groups and the Krull Topology
$\hat{\mathbb{Z}}$	the profinite completion of $\mathbb{Z}$, isomorphic to $\prod_p \mathbb{Z}_p$	Profinite Groups and the Krull Topology
Profinite completion $\hat{A}$	the inverse limit of the finite quotients of $A$	Profinite Groups and the Krull Topology
Inverse limit of finite abelian groups	a compact totally disconnected abelian group	Profinite Groups and the Krull Topology
$\mathbb{Q}_p$	the additive group of $p$-adic numbers; locally compact, not compact	The $p$-adic Numbers
Pontryagin dual $\hat{A} = \operatorname{Hom}(A,\mathbb{R}/\mathbb{Z})$	the dual of a locally compact abelian group; an exact contravariant equivalence	Pontryagin Duality (Part II)
Non-example: $\mathbb{Z}_p$	fails divisibility: $p$ does not divide $1$ in $\mathbb{Z}_p$	The $p$-adic Numbers
Non-example: $\mathbb{Q}$ with the discrete topology	fails compactness: its Pontryagin dual is not compact	Pontryagin Duality (Part II)

Abelian Groups as Modules and as Coefficients

Because an abelian group is exactly a module over $\mathbb{Z}$, the module theory of the corpus is the abelian theory in a general ring: the structure theorem over a principal ideal domain applies to $\mathbb{Z}$, the rank is the free rank, and the annihilator of an element is the ideal generated by its order. Abelian groups also serve as the coefficient objects of group cohomology and homology.

Object	The property it has	Introduced in
Abelian group as a $\mathbb{Z}$-module	the scalar action $na$ is the integer multiple	Modules
Structure theorem over a PID	the classification of finitely generated modules, hence of finitely generated abelian groups	Modules over a PID
Torsion submodule $M_{\mathrm{tor}}$	the module form of the torsion subgroup	Modules over a PID
Annihilator $\operatorname{Ann}(m)$	the ideal of scalars killing $m$, generated by $\operatorname{ord}(m)$ over $\mathbb{Z}$	Modules over a PID
Coefficient group of group cohomology	the abelian group $M$ in $H^n(G,M)$	Group Cohomology
Schur multiplier $H_2(G,\mathbb{Z})$	the second homology with abelian coefficients	Group Cohomology
Non-example: a non-free finitely generated torsion-free module over a general ring	fails the PID structure theorem: over a general ring it need not be free	Modules

The Classification in Each Family

Each family is pinned by one invariant, and the table gathers the four statements so that the family and its classifying theorem stand together.

Family	The classifying invariant	The theorem	Introduced in
Finitely generated	the rank and the invariant factors $d_1 \mid \cdots \mid d_k$	the structure theorem for finitely generated modules over a PID	Finitely Generated Abelian Groups
Torsion	the primary components, and the Ulm invariants within a countable primary group	Ulm's theorem	Infinite Abelian Groups
Torsion-free	the rank	rank classifies the free and the divisible cases, and fails in general	Infinite Abelian Groups
Divisible	the rank, with one copy of $\mathbb{Q}$ or $\mathbb{Z}[p^\infty]$ per rank	the structure theorem for divisible groups	Infinite Abelian Groups
Profinite	the Pontryagin dual, a discrete torsion group	Pontryagin duality	Pontryagin Duality (Part II)

Summary

The list gathers the abelian groups of the corpus. The finitely generated groups are the free abelian groups and the finite abelian groups, classified by the invariant factors and the elementary divisors; the torsion and torsion-free groups are separated by the torsion subgroup, the torsion groups decomposing into primary components and the countable primary groups classified by the Ulm invariants; the divisible groups are the direct sums of copies of $\mathbb{Q}$ and of the Prüfer groups, classified by their rank, with the divisible hull as the universal embedding; and the profinite abelian groups are the inverse limits of finite abelian groups, with $\mathbb{Z}_p$ and $\hat{\mathbb{Z}}$ as the standard examples and Pontryagin duality as their topological theory. The abelian groups are also the modules over $\mathbb{Z}$ and the coefficients of group cohomology. The non-examples — $\mathbb{Z}^r$, $\mathbb{Z}/n\mathbb{Z}$, the free abelian group of rank at least two, the countable torsion-free group of rank at least two, $\mathbb{Q}$ as a non-free group, $\mathbb{Z}$ and a finite group as non-divisible groups, $\mathbb{Z}_p$, and $\mathbb{Q}$ with the discrete topology — name the property that each fails.

Summary of Notation

The article denotes its objects by name; the symbols appearing in the tables are those of the introducing articles.

Symbol	Meaning
$\mathbb{Z}$, $\mathbb{Z}^r$, $\mathbb{Z}/n\mathbb{Z}$	integers, free abelian of rank $r$, cyclic group
$T(A)$, $A_p$, $A[p]$	torsion subgroup, primary component, $p$-torsion
$\operatorname{rk}(A)$	rank of an abelian group
$U(k,G)$	Ulm invariant of a countable primary group
$\mathbb{Q}$, $\mathbb{Q}/\mathbb{Z}$	rationals and the torsion divisible group
$\mathbb{Z}[p^\infty]$	Prüfer $p$-group
$D$, $D^{(I)}$	rank-one divisible group and its direct sum
$\mathbb{Z}_p$, $\hat{\mathbb{Z}}$, $\mathbb{Q}_p$	$p$-adic integers, profinite integers, $p$-adic numbers
$\operatorname{Hom}(A,\mathbb{R}/\mathbb{Z})$	Pontryagin dual
$\bigoplus_i A_i$, $\prod_i A_i$	direct sum and direct product
$d_1 \mid \cdots \mid d_k$, $p_i^{e_i}$	invariant factors and elementary divisors