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Divisible group

In group theory, a divisible group is an abelian group G such that for every element x \in G and every nonzero integer n, there exists an element y \in G satisfying ny = x.^[1]^[2] This property implies that every element is "divisible" by any integer, making divisible groups fundamental injective objects in the category of abelian groups.^[1] Key examples include the additive group of rational numbers \mathbb{Q}, which is divisible because for any q \in \mathbb{Q} and integer n \neq 0, q/n \in \mathbb{Q} satisfies the equation.^[2] Similarly, the additive group of real numbers \mathbb{R} is divisible, as is any vector space over \mathbb{Q} or \mathbb{R}.^[1] The Prüfer p-group \mathbb{Z}/p^\infty\mathbb{Z}, consisting of p-power roots of unity, is another divisible example for each prime p.^[1] In contrast, the integers \mathbb{Z} under addition are not divisible, as there is no integer y such that $2y = 1.^[2] No nontrivial finite abelian group is divisible.^[1] Divisible groups exhibit several important structural properties: the direct product and direct sum of divisible groups are divisible, and every quotient of a divisible group is divisible.^[1] They are precisely the injective \mathbb{Z}-modules, meaning any homomorphism from a subgroup of an abelian group to a divisible group extends to the whole group.^[1]^[2] Every abelian group G decomposes uniquely as G = dG \oplus R, where dG is the maximal divisible subgroup and R is reduced (containing no nontrivial divisible subgroups).^[1]^[2] By the structure theorem, every divisible group is a direct sum of copies of \mathbb{Q} (for the torsion-free part, forming a \mathbb{Q}-vector space) and copies of the Prüfer p-groups \mathbb{Z}/p^\infty\mathbb{Z} (for the p-primary torsion components, forming vector spaces over the field with p elements).^[1] This classification underscores their role in the study of abelian groups, homological algebra, and module theory over principal ideal domains like \mathbb{Z}.^[2]

Fundamentals

Definition

In the theory of abelian groups, which are commutative groups equipped with an addition operation satisfying the usual group axioms, a divisible group provides a fundamental notion of "divisibility" in an algebraic sense. An abelian group G is called divisible if, for every element g \in G and every positive integer n, there exists an element h \in G such that nh = g.^[3] This condition ensures that every element can be "divided" by any positive integer within the group itself, reflecting a form of algebraic completeness. Equivalent formulations of divisibility capture this property categorically or through homomorphisms. Specifically, G is divisible if and only if the multiplication-by-n map \mu_n: G \to G, defined by \mu_n(h) = nh, is surjective for every positive integer n > 0.^[3] Moreover, in the category of abelian groups (with the Hom functor), G is divisible if and only if it is an injective object, meaning that for any monomorphism A \hookrightarrow B of abelian groups, every homomorphism A \to G extends to a homomorphism B \to G; this equivalence follows from Baer's criterion applied to the category of abelian groups.^[3] The concept of divisible groups traces its origins to 19th-century developments in abstract algebra, where mathematicians like Leopold Kronecker introduced foundational ideas on abelian groups while studying number-theoretic extensions and decompositions. It was formalized in modern terms during the mid-20th century, notably by Irving Kaplansky in his systematic treatment of infinite abelian groups.^[3] This definition presupposes familiarity with basic abelian group theory and the associated category-theoretic framework.

Examples

The additive group of rational numbers (\mathbb{Q}, +) is a classical example of a divisible abelian group, as every rational can be divided by any nonzero integer to yield another rational. Another familiar instance is the circle group \mathbb{R}/\mathbb{Z}, which is divisible and isomorphic to the multiplicative group of complex numbers of modulus 1 (the unit circle S^1).^[4] The multiplicative group of nonzero complex numbers \mathbb{C}^* is also divisible, being isomorphic to the direct product \mathbb{R}_{>0} \times S^1, where both factors are divisible.^[4] The Prüfer p-group \mathbb{Z}(p^\infty), defined as the direct limit of the cyclic groups \mathbb{Z}/p^n\mathbb{Z} for n \geq 1, provides a torsion example of a divisible abelian group for each prime p.^[5] This group is countable, infinite, and every proper subgroup is finite cyclic.^[5] Direct sums of divisible groups are divisible; for instance, the direct sum of \kappa copies of \mathbb{Q}, denoted \bigoplus_{\kappa} \mathbb{Q}, is divisible for any cardinal \kappa.^[4] The torsion subgroup of \mathbb{Q}/\mathbb{Z}, which coincides with \mathbb{Q}/\mathbb{Z} itself, is another torsion divisible group, isomorphic to the direct sum \bigoplus_p \mathbb{Z}(p^\infty) over all primes p. In contrast, the additive group of integers \mathbb{[Z](/page/Z)} is not divisible, since the element 1 has no solution to $2x = [1](/page/1) within \mathbb{[Z](/page/Z)}.^[1]

Properties

Basic Properties

A fundamental property of divisible abelian groups is that every divisible subgroup of an abelian group is a direct summand.^[6]^[1] Specifically, if D is a divisible subgroup of an abelian group G, then there exists a subgroup K \leq G such that G = D \oplus K.^[7] Nontrivial divisible abelian groups are never finitely generated. This follows from the fact that finitely generated abelian groups decompose into a finite direct sum of cyclic groups, none of which (except the trivial group) satisfy the divisibility condition.^[1] The torsion subgroup of a divisible abelian group is itself divisible.^[1] For any divisible group D, if tD denotes its torsion subgroup, then for every element x \in tD and positive integer n, there exists y \in tD such that ny = x, since solutions exist in D and preserve torsion.^[1] Pure subgroups of divisible abelian groups are divisible.^[8] That is, if H is a pure subgroup of a divisible group D—meaning H \cap nD = nH for all positive integers n—then H satisfies the divisibility axiom.^[8] An abelian group G is divisible if and only if it has no nontrivial reduced quotients.^[1] Here, a reduced group is one with no nontrivial divisible subgroup; since quotients of divisible groups are divisible, any reduced quotient must be trivial, and conversely, the existence of a nontrivial reduced quotient (such as the reduced part in the unique decomposition G \cong D \oplus R) implies G is not divisible.^[1] This summand behavior extends to extensions: for a divisible group G, every short exact sequence of the form

$0 \to G \to E \to Q \to 0

splits, meaning E \cong G \oplus Q.^[6] For instance, the rational numbers \mathbb{Q} as a subgroup of any containing abelian group splits off as a direct summand.^[6]

Injective Nature

In the category of abelian groups, an abelian group G is an injective object if, for every injective homomorphism i: A \hookrightarrow B of abelian groups and every group homomorphism f: A \to G, there exists a group homomorphism g: B \to G such that g \circ i = f.^[9] This property means that the contravariant Hom functor \Hom(-, G) is exact, preserving exact sequences.^[10] Divisible abelian groups are precisely the injective objects in this category.^[9] To see this equivalence, Baer's criterion provides a characterization: an abelian group G is injective if and only if, for every positive integer n, every homomorphism \mathbb{Z}/n\mathbb{Z} \to G extends to a homomorphism \mathbb{Z} \to G.^[11] If G is divisible, then for any such map sending the generator of \mathbb{Z}/n\mathbb{Z} to some x \in G, there exists y \in G with n y = x, and extending by sending the generator of \mathbb{Z} to y works.^[12] Conversely, if G is not divisible, there exist x \in G and n > 0 such that no y satisfies n y = x; then the map \mathbb{Z}/n\mathbb{Z} \to G sending the generator to x cannot extend to \mathbb{Z} \to G, violating injectivity.^[12] A key consequence is the universal property that every abelian group embeds as a subgroup into some injective (hence divisible) abelian group, known as an injective hull.^[13] Explicitly, for any abelian group A, one can construct such an embedding into the divisible group (\mathbb{Q}/\mathbb{Z})^{\Hom(A, \mathbb{Q}/\mathbb{Z})} via the evaluation map a \mapsto (f \mapsto f(a))_{f \in \Hom(A, \mathbb{Q}/\mathbb{Z})}, which is injective because \mathbb{Q}/\mathbb{Z} is a cogenerator.^[13] The injective nature of divisible groups connects abelian group theory to broader homological algebra, where injectives facilitate resolutions and derived functors.^[10] This perspective originated with Reinhold Baer's 1940 introduction of injective modules as those embeddable as direct summands in every containing module, initially for abelian groups and later generalized to modules over rings.^[11]

Structure and Decomposition

Structure Theorem

The structure theorem for divisible abelian groups provides a complete classification up to isomorphism. Every divisible abelian group G decomposes as G \cong \Tor(G) \oplus \mathbb{Q}^{(I)}, where \Tor(G) is the torsion subgroup of G, isomorphic to a direct sum \bigoplus_p \mathbb{Z}(p^\infty)^{(\kappa_p)} over all primes p with cardinal invariants \kappa_p (the p-ranks), and \mathbb{Q}^{(I)} is a direct sum of |I| copies of the additive group of rational numbers \mathbb{Q}, forming a vector space over \mathbb{Q} of dimension |I|.^[14]^[15] This decomposition arises because divisible abelian groups admit a natural \mathbb{Q}-module structure, allowing scalar multiplication by rationals. The torsion-free quotient G / \Tor(G) is a torsion-free divisible \mathbb{Q}-module, hence free over \mathbb{Q} and isomorphic to \mathbb{Q}^{(I)} for some index set I. Separately, the torsion subgroup \Tor(G) is a divisible torsion group, which decomposes uniquely as a direct sum of Prüfer p-groups \mathbb{Z}(p^\infty) for each prime p, with the number of summands given by the cardinal \kappa_p. For instance, the group \mathbb{Q}/\mathbb{Z} exemplifies the torsion part as \bigoplus_p \mathbb{Z}(p^\infty), one copy per prime.^[14]^[15] The isomorphism type of a divisible abelian group is uniquely determined by its cardinal invariants: the rank |I| (dimension of the torsion-free part over \mathbb{Q}) and the p-ranks \kappa_p for each prime p (dimensions of the p-primary components). These invariants fully classify the group, as any two divisible groups with matching invariants are isomorphic.^[14]^[15]

Injective Envelope

The injective envelope of an abelian group G, denoted E(G), is the smallest injective (equivalently, divisible) abelian group containing G as an essential subgroup, where essential means that every nonzero subgroup of E(G) intersects G nontrivially.^[16] This construction leverages the fact that divisible groups are precisely the injective objects in the category of abelian groups.^[17] Every abelian group admits an injective envelope, which is unique up to isomorphism over G (that is, any two such envelopes are isomorphic via a map fixing G pointwise).^[16] The envelope is constructed as the divisible hull of G, the minimal divisible supergroup generated by G under the operation of "division" by integers, consisting of all formal fractions g/n for g \in G and positive integers n, modulo the relations ng/n = g.^[17] In general, E(G) = G + D for some divisible group D, where the sum denotes the subgroup generated by G and D. For torsion-free abelian groups G, the injective envelope E(G) coincides with the rational completion \mathbb{Q} \otimes_{\mathbb{Z}} G, which embeds G as a \mathbb{Z}-submodule of this \mathbb{Q}-vector space.^[18] A concrete example is G = \mathbb{Z}, whose injective envelope is E(\mathbb{Z}) = \mathbb{Q}, as \mathbb{Q} is the minimal divisible group containing \mathbb{Z} essentially.^[17]

Relation to Reduced Groups

In abelian group theory, a reduced group is defined as an abelian group R whose only divisible subgroup is the trivial subgroup \{0\}. This condition ensures that R contains no nontrivial divisible elements or structures. Equivalently, R is reduced if and only if \operatorname{Hom}(\mathbb{Q}, R) = 0, meaning there are no nontrivial homomorphisms from the rationals into R.^[19]^[20] A fundamental decomposition theorem states that every abelian group A can be uniquely expressed as a direct sum A = D \oplus R, where D is a divisible group serving as the maximal divisible subgroup of A, and R is reduced. The uniqueness follows from the fact that the maximal divisible subgroup D is characteristically determined as the intersection of all divisible supergroups or the sum of all divisible subgroups, and the complement R is then isomorphic to A/D, which inherits the reduced property since any divisible subgroup of A/D would correspond to a larger divisible subgroup in A. This decomposition highlights the separation of the "divisible core" from the "rigid" reduced remainder.^[21]^[22] Reduced groups lack any \mathbb{Q}-vector space structure, as the absence of \operatorname{Hom}(\mathbb{Q}, R) \neq 0 prevents R from supporting scalar multiplication by rationals in a way that would generate divisible elements. Representative examples include the integers \mathbb{Z}, which has no elements of infinite order divisible by all integers except multiples of itself, and the additive group of p-adic integers \mathbb{Z}_p for a prime p, a torsion-free group that is compact and thus cannot contain infinite divisible chains.^[20]^[23]

Generalizations and Extensions

To Modules

In the context of modules over a commutative ring R, the notion of divisibility extends the abelian group case, where \mathbb{Z}-modules coincide with abelian groups. For R an integral domain, an R-module M is divisible if multiplication by every nonzero element r \in R is surjective, i.e., rM = M for all $0 \neq r \in R.^[24] For general commutative rings, the definition requires this surjectivity only for regular elements r \in R (non-zero-divisors), ensuring compatibility with zero-divisors.^[25] Over a principal ideal domain like \mathbb{Z}, this reduces precisely to the divisible abelian group case, where modules such as \mathbb{Q} or \mathbb{Q}/\mathbb{Z} exemplify divisibility. Over a field, every module (i.e., vector space) is divisible, as multiplication by nonzero scalars is invertible. Over commutative Noetherian rings, every injective module is divisible, reflecting the extension property's compatibility with scalar multiplications. However, the converse does not hold in general; for instance, over \mathbb{Z}, the module \mathbb{Q}(x)/\mathbb{Z} is divisible but not injective.^[26] A generalization of Baer's criterion characterizes injectivity for R-modules by testing extensions over cyclic modules generated by ideals: M is injective if and only if every homomorphism from an ideal I \subseteq R to M extends to one from R to M. This aligns with divisibility testing in domains, where surjectivity over principal ideals suffices.

To Other Categories

In category theory, divisible groups arise as the injective objects within the category of abelian groups, denoted Ab, where an object is injective if every monomorphism into it extends over any larger domain. This equivalence holds because the Baer's criterion characterizes injectivity via divisibility by integers, and it generalizes to other abelian categories where injective objects enable resolutions for computing Ext and Tor functors in homological algebra.^[9]^[27] Beyond abelian groups, generalizations to non-abelian settings define a divisible group G such that for every x \in G and positive integer n, there exists y \in G satisfying y^n = x. Non-abelian examples are scarce compared to the abelian case and include unipotent subgroups of U_n(\mathbb{Q}), the group of upper triangular n \times n matrices over \mathbb{Q} with 1s on the diagonal, generated via the exponential map from Lie algebras of strictly upper triangular matrices. Another example is the multiplicative group of unit quaternions, which admits nth roots for every element. Gilbert Baumslag's foundational work on uniquely divisible groups—where roots are unique, analogous to vector spaces—explored their structure in non-abelian contexts, including free uniquely divisible groups and their relation to radicable subgroups in solvable groups.^[28] In other categories, such as Lie groups and topological groups, divisibility often incorporates continuity: a topological group is continuously divisible if nth roots exist via continuous paths. The additive group \mathbb{R} exemplifies a divisible abelian Lie group, as scalar multiplication by $1/n is continuous and surjective. More generally, connected simply connected Lie groups with surjective exponential maps, such as the universal cover of \mathrm{SL}(2, \mathbb{R}), are divisible, allowing elements to be expressed as continuous nth powers. Connected compact Lie groups, like \mathrm{SU}(2), also satisfy discrete divisibility, though continuous versions may fail for non-toral components. Recent developments link divisible structures to homological algebra in derived categories, where injective objects underpin t-structures and resolutions, though no major classification theorems for divisible groups have emerged since the 1970s. In algebraic geometry, the concept extends to injective sheaves on schemes, which are flasque and acyclic, facilitating injective resolutions for computing sheaf cohomology; for instance, the Godement resolution provides a canonical injective resolution for any sheaf of abelian groups. Applications appear in derived categories of coherent sheaves, aiding computations in étale or crystalline cohomology.^[29]^[30] A key limitation is the absence of a structure theorem for non-abelian divisible groups, unlike the abelian case where they decompose as direct sums of \mathbb{Q} and Prüfer p-groups; non-abelian examples resist such indecomposable classifications, with ongoing research focusing on specific constructions like those in unipotent varieties rather than global decompositions.

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