Infinite Abelian Groups

Introduction

An abelian group is a group whose operation is commutative, and the abelian theory is both richer and more rigid than the general theory. Every subgroup is normal, every quotient is again abelian, and the structure of a finitely generated abelian group is completely determined by two numerical invariants. For infinite abelian groups the classification is genuinely harder: torsion groups decompose into their primary components, divisible groups are classifiable completely, the countable primary groups are classified by the Ulm invariants, but the countable torsion-free groups are already too numerous to admit a comparably simple description.

This article is the tenth of the corpus and the first of the group articles. It stands below the whole foundational layer — Sets, Functions and Relations, Logic and Proof, Order Theory and Lattices, Cardinality and the Axiom of Choice, Set-Theoretic Foundations, Formal Logic and Computability, Model Theory, Proof Theory and Type Theory and Universal Properties and Categories — and it uses the elementary group theory of Groups, the direct and semidirect products, and the categorical vocabulary of universal properties. It develops the abelian theory for its own sake and because it is used later: the Schur multiplier and the low-dimensional cohomology take their coefficients in abelian groups, and the homology of a group is computed with abelian coefficients.

Because the article is the first of the group sequence, it also fixes a convention: abelian groups are written additively in this and the following group articles whenever the commutativity is essential, with $+$ the operation, $0$ the identity and $-a$ the inverse; a general group is written multiplicatively, as in Groups. The two notations are related by the dictionary $+ \leftrightarrow \cdot$, $0 \leftrightarrow e$, $-a \leftrightarrow a^{-1}$, $na \leftrightarrow a^n$ for $n \in \mathbb{Z}$. No distance, no topology and no form appears; the topological theory of abelian groups, including duality, belongs to Part II, and the module-theoretic reading of an abelian group is developed in the module articles, to which this one defers.

Elementary Theory of Abelian Groups

Additive Notation and Basic Facts

Definition. An abelian group is a group $(A, +)$ with $a + b = b + a$ for all $a,b \in A$. The identity is written $0$ and the inverse of $a$ is written $-a$; the order of an element $a$ is the least positive $n$ with $na = 0$, written $\operatorname{ord}(a)$, or $\infty$ if there is none, where

$$ na = \underbrace{a + a + \cdots + a}_{n} \quad (n > 0), \qquad 0a = 0, \qquad (-n)a = -(na). $$

The order of $A$ is the cardinality $|A|$.

Proposition. If $a \in A$ has finite order $n$, then the multiples of $a$ form the subgroup $\langle a\rangle = \{0, a, 2a, \ldots, (n-1)a\} \cong \mathbb{Z}/n\mathbb{Z}$. If $a$ and $b$ commute and have coprime finite orders $m$ and $n$, then $\operatorname{ord}(a+b) = mn$. In an abelian group $A$ of exponent $n$ — one in which $na = 0$ for all $a$ — the map $a \mapsto pa$ has kernel $A[p] = \{a : pa = 0\}$, the subgroup of elements of order dividing $p$, for each prime $p$.

Proof sketch. The first statement is the division algorithm in $\mathbb{Z}$ applied to the exponent of $a$. For the second, $mn(a+b) = nm\,a + mn\,b = 0$; if $k(a+b) = 0$ then $ka = -kb$ lies in $\langle a\rangle \cap \langle b\rangle$, which is trivial because $\gcd(m,n)=1$, so $m \mid k$ and $n \mid k$ and $mn \mid k$. The last statement is the definition of the kernel of an additive homomorphism.

Example. The groups $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$ are abelian, as are all direct sums of them and all subgroups and quotients of abelian groups; the group of all self-maps of a set with at least two elements, under composition, is not abelian, since two functions may fail to commute.

Cyclic Groups

Definition. A group is cyclic if it is generated by one element. A subgroup of a group $G$ is characteristic if it is invariant under every automorphism of $G$; every subgroup of an abelian group is normal, and the subgroups generated by the sets $\{a : na = 0\}$ and $\{na : a \in A\}$ are characteristic.

Theorem. Let $A$ be a cyclic group.

$A \cong \mathbb{Z}$ or $A \cong \mathbb{Z}/n\mathbb{Z}$ for some $n \geq 1$.
Every subgroup of $A$ is cyclic; the subgroups of $\mathbb{Z}/n\mathbb{Z}$ correspond to the divisors $d$ of $n$, the subgroup being generated by $n/d$.
The subgroups of $\mathbb{Z}$ are the $n\mathbb{Z}$ for $n \geq 0$.

Proof sketch. (1) If $A = \langle a\rangle$, the map $\mathbb{Z} \to A$, $k \mapsto ka$, is a surjective homomorphism, and its kernel is $n\mathbb{Z}$ for $n = \operatorname{ord}(a)$ (the subgroup of $\mathbb{Z}$ generated by the least positive element of the kernel); the first isomorphism theorem gives the conclusion. (2) If $B \leq A$ is nontrivial, let $d$ be the smallest positive integer with $da \in B$; the division algorithm shows that every element of $B$ is a multiple of $da$, so $B = \langle da\rangle \cong \mathbb{Z}/(n/d)$. (3) The subgroups of $\mathbb{Z}$ are the kernels of the homomorphisms to cyclic groups, that is, the $n\mathbb{Z}$.

Direct Sums

Definition. Let $(A_i)_{i \in I}$ be a family of abelian groups. Its direct sum $\bigoplus_{i \in I} A_i$ is the subgroup of the product $\prod_i A_i$ consisting of the families $(a_i)$ with $a_i = 0$ for all but finitely many $i$, with coordinatewise addition. When $I$ is finite the direct sum is the same as the product, and it is written $A_1 \oplus \cdots \oplus A_n$.

The direct sum is the coproduct in the category of abelian groups and the product is the categorical product; the two coincide for finitely many factors and differ for infinitely many, where the canonical map $\bigoplus_i A_i \to \prod_i A_i$ is injective but not surjective. The universal property, in the form used in Universal Properties and Categories, is that a homomorphism $\bigoplus_i A_i \to B$ is the same thing as an arbitrary family of homomorphisms $A_i \to B$, because every element of the direct sum involves only finitely many coordinates, and dually a homomorphism $B \to \prod_i A_i$ is the same thing as an arbitrary family of homomorphisms $B \to A_i$.

Proposition. Let $A$ be abelian and let $(A_i)$ be a family of subgroups. The following are equivalent: (i) the map $\bigoplus_i A_i \to A$, $(a_i) \mapsto \sum_i a_i$, is an isomorphism; (ii) every $a \in A$ is a sum of elements of the $A_i$ in exactly one way; (iii) $A$ is generated by the $A_i$ and $A_i \cap \sum_{j \neq i} A_j = 0$ for every $i$.

Proof. The equivalence of (i) and (ii) is the definition of isomorphism restricted to the finite-support families. For (ii) implies (iii), generation is clear and a nonzero element of the intersection would have two representations; the converse writes any element as a finite sum, using generation, and shows the representation unique by moving all but one term to the other side and applying the intersection condition.

Abelian Groups as Modules over $\mathbb{Z}$

The additive group of an abelian group carries a canonical action of the integers, $n \cdot a = na$, and this action satisfies

$$ (n + m)a = na + ma, \qquad n(a + b) = na + nb, \qquad (nm)a = n(ma), \qquad 1a = a. $$

These are the axioms of a module over the ring $\mathbb{Z}$, so that an abelian group is the same thing as a $\mathbb{Z}$-module and a homomorphism of abelian groups is the same thing as a $\mathbb{Z}$-linear map. The correspondence is used here as a dictionary rather than as a construction. The structure theory of finitely generated modules over a principal ideal domain is standard algebra, obtained from the Smith normal form of a presentation matrix (Lang, cited under Further Reading); it gives the structure theorem for finitely generated abelian groups as the special case of the base ring $\mathbb{Z}$, and the article quotes that proof where it is shorter than the group-theoretic one. The general theory of modules over a commutative ring is developed, in the Linear Spaces slot of this Part.

Free Abelian Groups and Rank

Free Abelian Groups and their Universal Property

Definition. An abelian group $F$ is free abelian with basis $B \subseteq F$ if every element of $F$ is a unique finite integer combination of elements of $B$. Equivalently, $F = \bigoplus_{b \in B} \mathbb{Z} b$ with $\mathbb{Z} b \cong \mathbb{Z}$. A free abelian group of finite basis cardinality $n$ is written $\mathbb{Z}^n$.

Theorem (universal property). Let $B$ be a set. There is a free abelian group $F(B)$ with basis $B$ such that for every abelian group $A$ and every function $f : B \to A$ there is a unique homomorphism $\tilde f : F(B) \to A$ extending $f$.

Proof. Take $F(B) = \bigoplus_{b \in B} \mathbb{Z}$ and let $B$ be the set of standard basis elements. Every element is a finite sum $\sum_b n_b b$, and the assignment $\tilde f(\sum_b n_b b) = \sum_b n_b f(b)$ is a homomorphism, unique because it is determined on the generators.

This is the universal arrow from a set to the forgetful functor, in the sense of Universal Properties and Categories, and it is the model of the free construction whose group-theoretic form is the free group. The free abelian group on $B$ is the abelianisation of the free group on $B$.

Rank

Definition. A subset $S$ of an abelian group $A$ is linearly independent if every relation $\sum_{s \in S} n_s s = 0$ with $n_s \in \mathbb{Z}$ and finitely many nonzero coefficients has all $n_s = 0$; a maximal linearly independent subset is a basis of the torsion-free part. The rank of $A$, written $\operatorname{rk}(A)$, is the cardinality of a maximal linearly independent subset.

Theorem. The rank of a free abelian group is well defined: if $A$ is free with basis $B$ and $A'$ is free with basis $B'$, and $A \cong A'$, then $|B| = |B'|$; in particular if $\mathbb{Z}^n \cong \mathbb{Z}^m$ then $n = m$. More generally the rank of an abelian group is well defined and equals the cardinality of a maximal linearly independent subset.

Proof. Let $A$ be free with basis $B$. The quotient $A/2A$ is a vector space over the field $\mathbb{F}_2$, and the canonical map exhibits it as the direct sum of one copy of $\mathbb{Z}/2\mathbb{Z}$ for each $b \in B$, so its dimension over $\mathbb{F}_2$ is $|B|$. Any two bases of a vector space have the same cardinality, by the standard exchange argument for the finite case and its well-ordering form in general; hence $|B|$ is determined by the isomorphism class of $A$, and it is recovered from $A/2A$ as $\log_2|A/2A|$ when $B$ is finite and as the cardinality $|A/2A|$ when $B$ is infinite. In particular $\mathbb{Z}^n \cong \mathbb{Z}^m$ with $n, m$ finite forces $2^n = |A/2A| = 2^m$ and $n = m$. For a general abelian group, a maximal linearly independent subset contains no torsion element, and the maximal linearly independent subsets of $A$ are exactly the preimages of those of the torsion-free quotient $A/T(A)$, so the rank of $A$ equals the rank of $A/T(A)$ and is well defined by the basis theorem in the torsion-free case.

Proposition. Let $A$ be an abelian group. A nonzero torsion element is linearly dependent on its own, since $\operatorname{ord}(t)\cdot t = 0$ is a nontrivial relation, so a linearly independent subset contains no torsion element; consequently the rank of $A$ equals the rank of the torsion-free quotient $A/T(A)$.

Subgroups of Free Abelian Groups

Theorem. Every subgroup of a free abelian group is free abelian, and if $F$ is free of rank $n$ and $H \leq F$ then $\operatorname{rk}(H) \leq n$.

Proof sketch. Let $B = \{b_1, \ldots, b_n\}$ be a basis. Induct on $n$. For $n = 1$, a subgroup of $\mathbb{Z}$ is $m\mathbb{Z}$, which is free of rank $1$ if $m \neq 0$ and rank $0$ if $m = 0$. For the induction step, let $\pi : F \to \mathbb{Z}$ be the projection onto the last coordinate and let $H' = H \cap \ker\pi$; by induction $H'$ is free of rank at most $n - 1$. If $\pi(H) = 0$ then $H = H'$. Otherwise $\pi(H) = m\mathbb{Z}$ with $m \neq 0$; choose $h \in H$ with $\pi(h) = m$ and show, by subtracting a multiple of $h$ from each element of $H$, that $H = H' \oplus \mathbb{Z} h$, which is free of rank at most $n$.

The theorem is the abelian case of the Nielsen–Schreier theorem for free groups, which is proved; for a free group the conclusion is that a subgroup of a free group is free, without the rank restriction, and rank can only increase. The induction above is written for a finite basis; for an arbitrary basis the same argument is run by transfinite induction along a well-ordered basis, so the theorem holds for free abelian groups of every rank, the well-ordering being the point at which the argument uses Cardinality and the Axiom of Choice.

Finitely Generated Abelian Groups

Theorem (structure theorem). Every finitely generated abelian group $A$ is isomorphic to a direct sum

$$ A \cong \mathbb{Z}^r \oplus \mathbb{Z}/d_1\mathbb{Z} \oplus \mathbb{Z}/d_2\mathbb{Z} \oplus \cdots \oplus \mathbb{Z}/d_k\mathbb{Z} $$

with $r \geq 0$, $d_i > 1$ and $d_1 \mid d_2 \mid \cdots \mid d_k$. The integer $r = \operatorname{rk}(A)$ and the sequence $(d_1,\ldots,d_k)$, the invariant factors, are uniquely determined by $A$. Equivalently $A$ is a direct sum of cyclic groups of prime-power order, the elementary divisors, uniquely determined up to order.

Proof sketch. As a $\mathbb{Z}$-module, $A$ is a finitely generated module over the principal ideal domain $\mathbb{Z}$; the standard structure theorem for such modules, obtained from the Smith normal form of a presentation matrix (Lang, under Further Reading), gives the decomposition and the uniqueness. The rank $r$ is the rank of the free part, and the $d_i$ are the invariant factors of the torsion part.

Example. The finitely generated abelian groups of order $16$ are, up to isomorphism, $\mathbb{Z}/16$, $\mathbb{Z}/8 \oplus \mathbb{Z}/2$, $\mathbb{Z}/4 \oplus \mathbb{Z}/4$, $\mathbb{Z}/4 \oplus \mathbb{Z}/2 \oplus \mathbb{Z}/2$ and $(\mathbb{Z}/2)^4$; there are five, one for each partition of $4$, and the count is verified in the computations accompanying this article. The finitely generated abelian groups as a class are treated separately from the infinite ones because their classification is complete and elementary; the infinite theory below concerns groups that are not finitely generated, and already the torsion groups of that class require the Ulm invariants.

Torsion and Torsion-Free Groups

The Torsion Subgroup

Definition. An element $a$ of an abelian group $A$ is a torsion element if $na = 0$ for some $n > 0$; the group is a torsion group if every element is torsion and torsion-free if the only torsion element is $0$. The set $T(A)$ of torsion elements is the torsion subgroup.

Proposition. $T(A)$ is a subgroup of $A$, every subgroup and quotient of a torsion group is a torsion group, and $A/T(A)$ is torsion-free.

Proof. If $ma = 0$ and $nb = 0$ then $mn(a+b) = 0$ and $m(-a) = 0$, so $T(A)$ is a subgroup. The statements about subgroups and quotients are immediate. If $a + T(A)$ had finite order $n$ then $na \in T(A)$, so $mna = 0$ for some $m$ and $a$ is torsion, hence $a \in T(A)$ and the class is zero.

Proposition. A torsion group is the direct sum of its subgroups $A_p = \{a : p^k a = 0 \text{ for some } k\}$, one for each prime $p$, and $A_p$ is the $p$-primary component.

Proof. Every element of finite order $n$ is a sum of elements of orders dividing the prime powers dividing $n$: writing $n = n_1 \cdots n_r$ as a product of coprime prime powers and $\sum_i u_i n/n_i = 1$ with integers $u_i$, the element $a$ equals $\sum_i u_i (n/n_i) a$, and each term $(n/n_i)a$ is annihilated by $n_i$. The decomposition is unique by the intersection condition of the direct-sum proposition.

The proposition is Prüfer's first theorem in the countable case; it says that the study of torsion abelian groups reduces to the study of $p$-groups for a single prime $p$, and it is why the classification below is stated for $p$-groups.

Torsion-Free Groups of Finite Rank and Types

Definition. For a torsion-free abelian group $A$ of rank $1$ and a prime $p$, the $p$-height of $a \neq 0$ is the largest $k$ with $a \in p^k A$, or $\infty$ if no such largest exists; the height sequence of $a$ records the heights at all primes, and the type of $A$ is the class of the height sequence of a nonzero element under the relation that identifies two height sequences when they agree at every prime outside a finite set, the finitely many exceptional values being finite. The condition on the exceptional values is essential: without it two rank-one groups that differ only in the height at a single prime, one finite and one infinite there, would have the same type although they are not isomorphic.

A rank-one torsion-free group need not admit solutions to the equations $nx = a$; its divisible hull is a smallest rank-one torsion-free group containing it in which every such equation has a solution, and all rank-one torsion-free groups with this property are isomorphic. We fix one such group once and for all and write it $D_1$: it is torsion-free of rank one, and $nx = a$ has a solution for every $a \in D_1$ and every $n > 0$. Its standard concrete model is the additive group of the fraction field of $\mathbb{Z}$, introduced in Localization and the Fraction Field in the category Rings and Fields; the section below uses only its group structure, and the field-theoretic identification is deferred to that article.

Theorem. Every rank-one torsion-free abelian group is isomorphic to a subgroup of $D_1$ containing a cyclic subgroup of infinite order, and two such groups are isomorphic if and only if they have the same type.

Proof sketch. Fix $a \neq 0$ in a rank-one torsion-free $A$ and fix a nonzero element $a_0 \in D_1$. For every $x \in A$ the elements $x$ and $a$ are linearly dependent, so there are integers $m, n$ with $n \neq 0$ and $nx = ma$. The assignment $x \mapsto (m/n)a_0$ is well defined because $A$ is torsion-free, and it is an injective homomorphism $A \to D_1$, so $A$ is identified with a subgroup of $D_1$. The type is an isomorphism invariant because an isomorphism preserves heights. Conversely the height sequence determines the subgroup up to isomorphism: the subgroup of $D_1$ generated by a cyclic group of infinite order together with, for each prime $p$, an element $y_p$ with $p^{k} y_p$ in that cyclic group for all $k$ up to the height at $p$, realises the given sequence, and two subgroups of $D_1$ with the same type differ by multiplication by a nonzero element, which is an automorphism of $D_1$.

Example. The types of a subgroup of $D_1$ generated by a nonzero element, of the subgroup $H_2$ of elements divisible by every power of $2$, and of $D_1$ itself are distinct; the first has all heights finite at every prime, $H_2$ has infinite height at $2$ and finite height at the odd primes, and $D_1$ has infinite height at every prime. The classification of rank-one torsion-free groups by types is due to Baer, and it is the only case of the torsion-free classification that is complete in this sense; rank-two torsion-free groups already fail to be classifiable by a comparable invariant.

Divisible Groups

Definition and Examples

Definition. An abelian group $A$ is divisible if for every $a \in A$ and every positive integer $n$ the equation $nx = a$ has a solution $x \in A$; equivalently $nA = A$ for every $n > 0$.

Proposition. Every quotient of a divisible group is divisible; a direct sum of divisible groups is divisible; a subgroup of a divisible group need not be divisible (for instance the subgroup generated by a nonzero element of $D_1$ is infinite cyclic and not divisible); and a divisible subgroup of any abelian group is a direct summand.

Proof sketch. The quotient statement is the equation $n(A/B) = (nA + B)/B = A/B$. For the summand statement, let $D \leq A$ be divisible and choose by Zorn's lemma a subgroup $C$ maximal subject to $C \cap D = 0$. If $D + C \neq A$, take $a \notin D + C$. If the class of $a$ in $A/(D+C)$ has infinite order, then $ka \notin D + C$ for every $k \neq 0$, so $C + \langle a\rangle$ still meets $D$ trivially, contradicting maximality. If the class has finite order $n$, then $na = c + d$ with $c \in C$, $d \in D$; since $D$ is divisible, write $d = n d_0$ with $d_0 \in D$ and set $b = a - d_0$, so that $nb = c \in C$ while $b \notin D + C$. Then $C + \langle b\rangle$ meets $D$ trivially: an element $c' + kb \in D$ with $0 < k < n$ gives $kb \in D + C$, and if $d$ is the order of $b$ modulo $D + C$, which divides $n$ because $nb \in C$, then $d \mid k$ and $db = \delta + c''$ with $\delta \in D$, $c'' \in C$; since $db \notin C$, the element $\delta = db - c''$ is a nonzero element of $(C + \langle b\rangle) \cap D$, contradicting the maximality of $C$. Hence $D + C = A$.

Proposition. The group $D_1$ is divisible; a torsion-free divisible group is a direct sum of copies of $D_1$; a finite abelian group is divisible only if it is trivial.

Proof sketch. The divisibility of $D_1$ is part of its definition. If $A$ is torsion-free and divisible, let $(a_i)$ be a maximal linearly independent family, chosen by Zorn's lemma; each $a_i$ generates an infinite cyclic subgroup whose divisible hull inside $A$ is a copy of $D_1$, these copies meet trivially, and their sum is all of $A$, because maximality gives $mx = \sum_i n_i a_i$ with $m \neq 0$ for every $x \in A$ and divisibility divides the $a_i$ by $m$. A finite group has a nonzero element of maximal order and cannot be divisible.

The Prüfer Groups

Definition. For a prime $p$, the Prüfer group $\mathbb{Z}[p^\infty]$ is the group presented by

$$ \mathbb{Z}[p^\infty] = \langle x_1, x_2, x_3, \ldots \mid p\,x_1 = 0,\ p\,x_{k+1} = x_k \text{ for } k \geq 1\rangle. $$

Proposition. $\mathbb{Z}[p^\infty]$ is divisible, its elements of order dividing $p^k$ form a cyclic subgroup of order $p^k$, and every proper subgroup of $\mathbb{Z}[p^\infty]$ is finite and cyclic. The subgroups of order $p^k$ form a chain, and the quotient of $\mathbb{Z}[p^\infty]$ by the subgroup of order $p^k$ is again a Prüfer group.

Proof sketch. Write $\mathbb{Z}[p^\infty]$ as the increasing union of the cyclic groups $C_k = \langle x_k\rangle \cong \mathbb{Z}/p^k$, so that $C_k$ is exactly the set of elements of order dividing $p^k$. Divisibility: given $x$ of order $p^k$ and $n = p^j m$ with $p \nmid m$, solve $p^j y = x$ inside $C_{k+j}$ and then $m z = y$ inside $C_{k+j}$, using that multiplication by $m$ is invertible on a $p$-group; then $nz = x$. A proper subgroup is contained in some $C_k$, because a subgroup containing elements of unbounded order would contain every $C_k$; subgroups of a cyclic $p$-group are the chain of the $C_j$.

The verification accompanying this article constructs $\mathbb{Z}[p^\infty]$ as the increasing union of the layers $C_k$ and checks the divisibility and the lattice of subgroups for $p = 2$ and $p = 3$.

The Classification of Divisible Groups

Theorem. Every divisible abelian group is a direct sum of copies of $D_1$ and of copies of the groups $\mathbb{Z}[p^\infty]$, one family for each prime $p$:

$$ D \cong D_1^{(I)} \oplus \bigoplus_p \mathbb{Z}[p^\infty]^{(I_p)}. $$

The cardinal numbers $|I|$ and $|I_p|$ are uniquely determined by $D$.

Proof sketch. Let $T = T(D)$ be the torsion subgroup, which is divisible: if $t \in T$ has order $n$ and $nx = t$, then $n^2x = nt = 0$, so $x \in T$. The torsion divisible group $T$ splits into primary components $T_p$, each of which is divisible and is a direct sum of copies of $\mathbb{Z}[p^\infty]$: in a divisible $p$-group, the subgroup of elements of order dividing $p$ is elementary abelian and the group is the direct sum of the Prüfer groups generated by choosing a basis of that layer and lifting it along the division. The quotient $D/T$ is torsion-free and divisible, hence a direct sum of copies of $D_1$ by the proposition above. Uniqueness follows because $D[p] = \mathbb{Z}[p^\infty][p]^{(I_p)}$ is an elementary abelian $p$-group of dimension $|I_p|$ over $\mathbb{F}_p$, so $|I_p|$ is recovered from $D$ as that dimension, and $|I|$ is recovered as the rank of the torsion-free divisible quotient $D/T$.

The theorem classifies the divisible groups completely, and with it the injective abelian groups: an abelian group is injective (a direct summand of every group containing it) if and only if it is divisible, by Baer's criterion. The injective objects of a module category are treated; the group-theoretic statement is recorded here because it is the origin of the notion.

The Structure of Infinite Abelian Groups

Direct Sums of Cyclic Groups

Definition. An abelian group is a direct sum of cyclic groups if it is isomorphic to $\bigoplus_{i \in I} C_i$ with each $C_i$ cyclic. A torsion group of this form is a direct sum of finite cyclic groups, and the nonzero finite cyclic summands are unique up to order.

Theorem (Prüfer's second theorem). A countable torsion abelian group is a direct sum of cyclic groups if and only if none of its primary components has a nonzero element of infinite height, that is, an element divisible by every power of $p$. In that case the group is a direct sum of finite cyclic groups.

Proof sketch. If $A$ is the direct sum of the finite cyclic groups $C_i$ then an element of $A$ has $p$-height at most the largest exponent occurring among the orders of the $C_i$ that are $p$-groups, so no nonzero element is divisible by every power of $p$. Conversely, if a countable $p$-group has no nonzero element of infinite height, one chooses elements of maximal order in successive quotients by the subgroups already removed and shows that the resulting family of cyclic subgroups generates $A$ directly; the countability is used to run the selection transfinitely.

Ulm's Theorem

Definition. Let $G$ be an abelian $p$-group. For $k \geq 0$ let $G[p] = \{x : px = 0\}$, and define the $k$-th Ulm invariant by

$$ U(k, G) = \dim_{\mathbb{F}_p}\!\big((p^k G)[p] \,/\, (p^{k+1} G)[p]\big), $$

where $p^k G = \{p^k x : x \in G\}$ and $(p^k G)[p]$ is the subgroup of its elements of order dividing $p$. Each $U(k,G)$ is a cardinal number, finite when $G$ is finite, and the Ulm sequence of $G$ is $(U(0,G), U(1,G), U(2,G), \ldots)$.

Theorem (Ulm). Two countable abelian $p$-groups are isomorphic if and only if they have the same Ulm sequence. Every sequence of cardinal numbers $\kappa_0, \kappa_1, \kappa_2, \ldots$ with $\kappa_k \leq \aleph_0$ for every $k$ arises as the Ulm sequence of a countable $p$-group, namely of the direct sum of $\kappa_k$ copies of $\mathbb{Z}/p^{k+1}\mathbb{Z}$ over all $k$; a sequence with some $\kappa_k$ uncountable is the Ulm sequence of no countable group, and it is realised by an uncountable direct sum.

Proof sketch. The invariants are isomorphism-invariant because an isomorphism carries $p^k G$ to $p^k G'$ and $G[p]$ to $G'[p]$. For the converse, one uses the Ulm factors $U(k,G) = (p^kG)[p]/(p^{k+1}G)[p]$ and builds an isomorphism between two countable groups with the same sequence by a back-and-forth argument: at each stage one matches a chosen element of $G$ with an element of $G'$ of the same height profile relative to the elements already matched, using that the Ulm invariant counts the elements of each height modulo those of larger height. The argument is the standard transfinite back-and-forth of countable algebra and is given in the references.

Example. The finite abelian $2$-groups $\mathbb{Z}/8$, $\mathbb{Z}/4 \oplus \mathbb{Z}/2$ and $(\mathbb{Z}/2)^3$ have Ulm sequences $(0,0,1,0,\ldots)$, $(1,1,0,\ldots)$ and $(3,0,\ldots)$ respectively, so the theorem distinguishes them; the verification for this article computes the invariants from the definition for all partitions of $4$.

Ulm's theorem is the classification of countable primary groups, and it is the deepest positive result on the structure of infinite abelian groups. It is genuinely a classification by a complete invariant, in the sense in which the rank and the invariant factors classify the finitely generated groups.

Beyond the Countable Case

For uncountable $p$-groups the Ulm sequence no longer classifies, and the isomorphism problem for the torsion groups of cardinality $\aleph_1$ is not settled by the invariants of Ulm's theorem; the finer invariants of Shelah and the independence phenomena they exhibit are the subject of set-theoretic group theory and belong to the model-theoretic study of the category of abelian groups. Even in the countable case, the torsion-free groups are not classifiable: the rank-one groups are classified by types, but rank-two torsion-free groups already include families that are not separated by any reasonable complete invariant, and the classification problem for countable torsion-free abelian groups is undecidable in the sense that the isomorphism relation is not smooth. The contrast — a complete classification for the primary groups and none for the torsion-free groups — is the central fact of the structure theory of infinite abelian groups.

Remark. The dual theory, in which a group carries a topology and the group of characters into the circle group of Part II is studied with its own structure, is Pontryagin duality, and it belongs to Part II, where a topology is available; the algebraic part of the theory, in which the character group is treated as an abstract group, is developed with the module theory. The present article stops at the algebraic classification.

Summary

An abelian group is written additively; every subgroup is normal, the cyclic groups are $\mathbb{Z}$ and $\mathbb{Z}/n\mathbb{Z}$, and their subgroups are cyclic with the divisors of $n$ indexing those of $\mathbb{Z}/n\mathbb{Z}$. Direct sums are the coproducts , for finitely many factors, the products; an abelian group is the same thing as a module over $\mathbb{Z}$, and this dictionary transfers the module-theoretic structure theory, proved, to the finitely generated case.

Free abelian groups are the direct sums of copies of $\mathbb{Z}$, characterised by the universal property of extension of functions on a basis; the rank is well defined and is computed by reduction modulo $2$, and subgroups of free abelian groups are free of no larger rank. Every finitely generated abelian group is $\mathbb{Z}^r$ plus a direct sum of cyclic groups of prime-power order, with invariants the rank $r$ and the invariant factors, uniquely determined.

The torsion subgroup of an abelian group is the set of elements of finite order, the quotient by it is torsion-free, and a torsion group is the direct sum of its primary components, one for each prime. In rank one, torsion-free groups are the subgroups of the rank-one divisible torsion-free group $D_1$, classified by their types, and this is the only complete torsion-free classification. Divisible groups satisfy $nA = A$; every divisible group is a direct sum of copies of $D_1$ and of Prüfer groups $\mathbb{Z}[p^\infty]$, and the multiplicities are unique; the divisible groups are exactly the injective abelian groups.

For countable primary groups the structure theory is complete: a countable $p$-group is a direct sum of cyclic groups exactly when it has no nonzero element of infinite height (Prüfer's second theorem), and Ulm's theorem classifies the countable $p$-groups by their Ulm sequence, a sequence of cardinal numbers computed from the layers $(p^kG)[p]$. Beyond the countable case the invariants fail to classify, and the torsion-free groups of rank at least two are not classifiable by a complete invariant; the topological dual theory belongs to Part II.

Summary of Notation

Symbol	Meaning
$(A,+)$, $0$, $-a$, $na$	Abelian group in additive notation; identity; inverse; integer multiple
$\operatorname{ord}(a)$	Order of an element
$\mathbb{Z}/n\mathbb{Z}$	Cyclic group of order $n$
$A[p]$	Elements of order dividing $p$
$\bigoplus_i A_i$, $\prod_i A_i$	Direct sum and direct product of abelian groups
$T(A)$	Torsion subgroup
$A_p$	$p$-primary component
$F(B)$, $\mathbb{Z}^n$	Free abelian group on a set; free abelian group of rank $n$
$\operatorname{rk}(A)$	Rank
$D_1$	Fixed rank-one divisible torsion-free group; divisible hull of an infinite cyclic group
$p^k A$, $p$-height	Subgroup of $p^k$-th multiples; largest $k$ with $a \in p^kA$
$\mathbb{Z}[p^\infty]$	Prüfer $p$-group
$U(k,G)$	$k$-th Ulm invariant of a $p$-group
$D^{(I)}$	Direct sum of $\|I\|$ copies of $D$