# Divisible group

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Abelian group in which every element can, in some sense, be divided by positive integers

In [mathematics](/source/Mathematics), specifically in the field of [group theory](/source/Group_theory), a **divisible group** is an [abelian group](/source/Abelian_group) in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an *n*th multiple for each positive integer *n*. Divisible groups are important in understanding the structure of abelian groups, especially because they are the [injective](/source/Injective_module) abelian groups.

## Definition

An abelian group ( G , + ) {\displaystyle (G,+)} is **divisible** if, for every positive integer n {\displaystyle n} and every g ∈ G {\displaystyle g\in G} , there exists y ∈ G {\displaystyle y\in G} such that n y = g {\displaystyle ny=g} .[1] An equivalent condition is: for any positive integer n {\displaystyle n} , n G = G {\displaystyle nG=G} , since the existence of y {\displaystyle y} for every n {\displaystyle n} and g {\displaystyle g} implies that n G ⊇ G {\displaystyle nG\supseteq G} , and the other direction n G ⊆ G {\displaystyle nG\subseteq G} is true for every group. A third equivalent condition is that an abelian group G {\displaystyle G} is divisible if and only if G {\displaystyle G} is an [injective object](/source/Injective_object) in the [category of abelian groups](/source/Category_of_abelian_groups); for this reason, a divisible group is sometimes called an **injective group**.

An abelian group is p {\displaystyle p} -**divisible** for a [prime](/source/Prime_number) p {\displaystyle p} if for every g ∈ G {\displaystyle g\in G} , there exists y ∈ G {\displaystyle y\in G} such that p y = g {\displaystyle py=g} . Equivalently, an abelian group is p {\displaystyle p} -divisible if and only if p G = G {\displaystyle pG=G} .

## Examples

- The [rational numbers](/source/Rational_number) Q {\displaystyle \mathbb {Q} } form a divisible group under addition.

- More generally, the underlying additive group of any [vector space](/source/Vector_space) over Q {\displaystyle \mathbb {Q} } is divisible.

- Every [quotient](/source/Quotient_group) of a divisible group is divisible. Thus, Q / Z {\displaystyle \mathbb {Q} /\mathbb {Z} } is divisible.

- The *p*-[primary component](/source/Primary_component) Z [ 1 / p ] / Z {\displaystyle \mathbb {Z} [1/p]/\mathbb {Z} } of Q / Z {\displaystyle \mathbb {Q} /\mathbb {Z} } , which is [isomorphic](/source/Group_isomorphism) to the *p*-[quasicyclic group](/source/Quasicyclic_group) Z [ p ∞ ] {\displaystyle \mathbb {Z} [p^{\infty }]} , is divisible.

- The multiplicative group of the [complex numbers](/source/Complex_number) C ∗ {\displaystyle \mathbb {C} ^{*}} is divisible.

- Every [existentially closed](/source/Existentially_closed) abelian group (in the [model theoretic](/source/Model_theory) sense) is divisible.

## Properties

- If a divisible group is a [subgroup](/source/Subgroup) of an abelian group then it is a [direct summand](/source/Direct_summand) of that abelian group.[2]

- Every abelian group can be [embedded](/source/Embedding) in a divisible group.[3] Put another way, the category of abelian groups [has enough injectives](/source/Injective_object#Enough_injectives_and_injective_hulls).

- Non-trivial divisible groups are not [finitely generated](/source/Finitely_generated_abelian_group).

- Further, every abelian group can be embedded in a divisible group as an [essential subgroup](/source/Essential_subgroup) in a unique way.[4]

- An abelian group is divisible if and only if it is *p*-divisible for every prime *p*.

- Let A {\displaystyle A} be a [ring](/source/Ring_(mathematics)). If T {\displaystyle T} is a divisible group, then H o m Z -Mod ( A , T ) {\displaystyle \mathrm {Hom} _{\mathbf {Z} {\text{-Mod}}}(A,T)} is injective in the [category](/source/Category_(mathematics)) of A {\displaystyle A} -[modules](/source/Module_(mathematics)).[5]

## Structure theorem of divisible groups

Let *G* be a divisible group. Then the [torsion subgroup](/source/Torsion_subgroup) Tor(*G*) of *G* is divisible. Since a divisible group is an [injective module](/source/Injective_module), Tor(*G*) is a [direct summand](/source/Direct_summand) of *G*. So

- G = T o r ( G ) ⊕ G / T o r ( G ) . {\displaystyle G=\mathrm {Tor} (G)\oplus G/\mathrm {Tor} (G).}

As a quotient of a divisible group, *G*/Tor(*G*) is divisible. Moreover, it is [torsion-free](/source/Torsion_(algebra)). Thus, it is a vector space over **Q** and so there exists a set *I* such that

- G / T o r ( G ) = ⨁ i ∈ I Q = Q ( I ) . {\displaystyle G/\mathrm {Tor} (G)=\bigoplus _{i\in I}\mathbb {Q} =\mathbb {Q} ^{(I)}.}

The structure of the torsion subgroup is harder to determine, but one can show[6][7] that for all [prime numbers](/source/Prime_number) *p* there exists I p {\displaystyle I_{p}} such that

- ( T o r ( G ) ) p = ⨁ i ∈ I p Z [ p ∞ ] = Z [ p ∞ ] ( I p ) , {\displaystyle (\mathrm {Tor} (G))_{p}=\bigoplus _{i\in I_{p}}\mathbb {Z} [p^{\infty }]=\mathbb {Z} [p^{\infty }]^{(I_{p})},}

where ( T o r ( G ) ) p {\displaystyle (\mathrm {Tor} (G))_{p}} is the *p*-primary component of Tor(*G*).

Thus, if **P** is the set of prime numbers,

- G = ( ⨁ p ∈ P Z [ p ∞ ] ( I p ) ) ⊕ Q ( I ) . {\displaystyle G=\left(\bigoplus _{p\in \mathbf {P} }\mathbb {Z} [p^{\infty }]^{(I_{p})}\right)\oplus \mathbb {Q} ^{(I)}.}

The cardinalities of the sets *I* and *I**p* for *p* ∈ **P** are uniquely determined by the group *G*.

## Injective envelope

Main article: [Injective envelope](/source/Injective_envelope)

As stated above, any abelian group *A* can be uniquely embedded in a divisible group *D* as an [essential subgroup](/source/Essential_subgroup). This divisible group *D* is the **injective envelope** of *A*, and this concept is the [injective hull](/source/Injective_hull) in the category of abelian groups.

## Reduced abelian groups

An abelian group is said to be **reduced** if its only divisible subgroup is {0}. Every abelian group is the direct sum of a divisible subgroup and a reduced subgroup. In fact, there is a unique largest divisible subgroup of any group, and this divisible subgroup is a direct summand.[8] This is a special feature of [hereditary rings](/source/Hereditary_ring) like the integers **Z**: the [direct sum](/source/Direct_sum_of_modules) of injective modules is injective because the ring is [Noetherian](/source/Noetherian_ring), and the quotients of injectives are injective because the ring is hereditary, so any submodule generated by injective modules is injective. The converse is a result of ([Matlis 1958](#CITEREFMatlis1958)): if every module has a unique maximal injective submodule, then the ring is hereditary.

A complete classification of countable reduced periodic abelian groups is given by [Ulm's theorem](/source/Ulm's_theorem).

## Generalization

Several distinct definitions generalize divisible groups to divisible modules. The following definitions have been used in the literature to define a **divisible [module](/source/Module_(mathematics))** *M* over a [ring](/source/Ring_(mathematics)) *R*:

1. *rM* = *M* for all nonzero *r* in *R*.[9] (It is sometimes required that *r* is not a zero-divisor, and some authors[10] require that *R* is a [domain](/source/Domain_(ring_theory)).)

1. For every principal left [ideal](/source/Ideal_(ring_theory)) *Ra*, any [homomorphism](/source/Module_homomorphism) from *Ra* into *M* extends to a homomorphism from *R* into *M*.[11][12] (This type of divisible module is also called *principally injective module*.)

1. For every [finitely generated](/source/Finitely_generated_module) left ideal *L* of *R*, any homomorphism from *L* into *M* extends to a homomorphism from *R* into *M*.[*[citation needed](https://en.wikipedia.org/wiki/Wikipedia:Citation_needed)*]

The last two conditions are "restricted versions" of the [Baer's criterion](/source/Baer's_criterion) for [injective modules](/source/Injective_module). Since injective left modules extend homomorphisms from *all* left ideals to *R*, injective modules are clearly divisible in sense 2 and 3.

If *R* is additionally a domain then all three definitions coincide. If *R* is a principal left ideal domain, then divisible modules coincide with injective modules.[13] Thus in the case of the ring of integers **Z**, which is a principal ideal domain, a **Z**-module (which is exactly an abelian group) is divisible if and only if it is injective.

If *R* is a [commutative](/source/Commutative_ring) domain, then the injective *R* modules coincide with the divisible *R* modules if and only if *R* is a [Dedekind domain](/source/Dedekind_domain).[13]

## See also

- [Injective object](/source/Injective_object)

- [Injective module](/source/Injective_module)

- [Pure subgroup](/source/Pure_subgroup)

## Notes

1. **[^](#cite_ref-1)** Griffith, p.6

1. **[^](#cite_ref-2)** Hall, p.197

1. **[^](#cite_ref-3)** Griffith, p.17

1. **[^](#cite_ref-4)** Griffith, p.19

1. **[^](#cite_ref-5)** Lang, p. 106

1. **[^](#cite_ref-FOOTNOTEKaplansky1965_6-0)** [Kaplansky 1965](#CITEREFKaplansky1965).

1. **[^](#cite_ref-FOOTNOTEFuchs1970_7-0)** [Fuchs 1970](#CITEREFFuchs1970).

1. **[^](#cite_ref-8)** Griffith, p.7

1. **[^](#cite_ref-FOOTNOTEFeigelstock2006_9-0)** [Feigelstock 2006](#CITEREFFeigelstock2006).

1. **[^](#cite_ref-FOOTNOTECartanEilenberg1999_10-0)** [Cartan & Eilenberg 1999](#CITEREFCartanEilenberg1999).

1. **[^](#cite_ref-FOOTNOTELam1999_11-0)** [Lam 1999](#CITEREFLam1999).

1. **[^](#cite_ref-FOOTNOTENicholsonYousif2003_12-0)** [Nicholson & Yousif 2003](#CITEREFNicholsonYousif2003).

1. ^ [***a***](#cite_ref-FOOTNOTELam1999p.70—73_13-0) [***b***](#cite_ref-FOOTNOTELam1999p.70—73_13-1) [Lam 1999](#CITEREFLam1999), p.70—73.

## References

- [Cartan, Henri](/source/Henri_Cartan); [Eilenberg, Samuel](/source/Samuel_Eilenberg) (1999), *Homological algebra*, Princeton Landmarks in Mathematics, Princeton, NJ: Princeton University Press, pp. xvi+390, [ISBN](/source/ISBN_(identifier)) [0-691-04991-2](https://en.wikipedia.org/wiki/Special:BookSources/0-691-04991-2), [MR](/source/MR_(identifier)) [1731415](https://mathscinet.ams.org/mathscinet-getitem?mr=1731415) With an appendix by David A. Buchsbaum; Reprint of the 1956 original

- Feigelstock, Shalom (2006), "Divisible is injective", *Soochow J. Math.*, **32** (2): 241–243, [ISSN](/source/ISSN_(identifier)) [0250-3255](https://search.worldcat.org/issn/0250-3255), [MR](/source/MR_(identifier)) [2238765](https://mathscinet.ams.org/mathscinet-getitem?mr=2238765)

- Griffith, Phillip A. (1970). *Infinite Abelian group theory*. Chicago Lectures in Mathematics. University of Chicago Press. [ISBN](/source/ISBN_(identifier)) [0-226-30870-7](https://en.wikipedia.org/wiki/Special:BookSources/0-226-30870-7).

- [Hall, Marshall Jr](/source/Marshall_Hall_(mathematician)) (1959). *The theory of groups*. New York: Macmillan. Chapter 13.3.

- [Kaplansky, Irving](/source/Irving_Kaplansky) (1965). *Infinite Abelian Groups*. University of Michigan Press.

- [Fuchs, László](/source/L%C3%A1szl%C3%B3_Fuchs) (1970). *Infinite Abelian Groups Vol 1*. Academic Press.

- [Lam, Tsit-Yuen](/source/Tsit_Yuen_Lam) (1999), *Lectures on modules and rings*, Graduate Texts in Mathematics No. 189, vol. 189, Berlin, New York: [Springer-Verlag](/source/Springer-Verlag), [doi](/source/Doi_(identifier)):[10.1007/978-1-4612-0525-8](https://doi.org/10.1007%2F978-1-4612-0525-8), [ISBN](/source/ISBN_(identifier)) [978-0-387-98428-5](https://en.wikipedia.org/wiki/Special:BookSources/978-0-387-98428-5), [MR](/source/MR_(identifier)) [1653294](https://mathscinet.ams.org/mathscinet-getitem?mr=1653294)

- [Serge Lang](/source/Serge_Lang) (1984). *Algebra, Second Edition*. Menlo Park, California: Addison-Wesley.

- Matlis, Eben (1958). ["Injective modules over Noetherian rings"](http://projecteuclid.org/getRecord?id=euclid.pjm/1103039896). *Pacific Journal of Mathematics*. **8** (3): 511–528. [doi](/source/Doi_(identifier)):[10.2140/pjm.1958.8.511](https://doi.org/10.2140%2Fpjm.1958.8.511). [ISSN](/source/ISSN_(identifier)) [0030-8730](https://search.worldcat.org/issn/0030-8730). [MR](/source/MR_(identifier)) [0099360](https://mathscinet.ams.org/mathscinet-getitem?mr=0099360).

- Nicholson, W. K.; Yousif, M. F. (2003), *Quasi-Frobenius rings*, Cambridge Tracts in Mathematics, vol. 158, Cambridge: Cambridge University Press, pp. xviii+307, [doi](/source/Doi_(identifier)):[10.1017/CBO9780511546525](https://doi.org/10.1017%2FCBO9780511546525), [ISBN](/source/ISBN_(identifier)) [0-521-81593-2](https://en.wikipedia.org/wiki/Special:BookSources/0-521-81593-2), [MR](/source/MR_(identifier)) [2003785](https://mathscinet.ams.org/mathscinet-getitem?mr=2003785)

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