Editing Divisible group

{{Short description|Abelian group in which every element can, in some sense, be divided by positive integers}}
In [[mathematics]], specifically in the field of [[group theory]], a '''divisible group''' is an [[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 module|injective]] abelian groups.

==Definition==

An abelian group <math>(G, +)</math> is '''divisible''' if, for every positive integer <math>n</math> and every <math>g \in G</math>, there exists <math>y \in G</math> such that <math>ny=g</math>.<ref>Griffith, p.6</ref>  An equivalent condition is:  for any  positive integer <math>n</math>, <math>nG=G</math>, since the existence of <math>y</math> for every <math>n</math> and <math>g</math> implies that <math>n G\supseteq G</math>, and the other direction <math>n G\subseteq G</math> is true for every group. A third equivalent condition is that an abelian group <math>G</math> is divisible if and only if <math>G</math> is an [[injective object]] in the [[category of abelian groups]]; for this reason, a divisible group is sometimes called an '''injective group'''.

An abelian group is <math>p</math>-'''divisible''' for a [[prime number|prime]] <math>p</math> if for every <math>g \in G</math>, there exists <math>y \in G</math> such that <math>py=g</math>.  Equivalently, an abelian group is <math>p</math>-divisible if and only if <math>pG=G</math>.

==Examples==

* The [[rational number]]s <math>\mathbb Q</math> form a divisible group under addition.
* More generally, the underlying additive group of any [[vector space]] over <math>\mathbb Q</math> is divisible.
* Every [[quotient group|quotient]] of a divisible group is divisible. Thus, <math>\mathbb Q/\mathbb Z</math> is divisible.
* The ''p''-[[primary component]] <math>\mathbb Z[1/p]/\mathbb Z</math> of <math>\mathbb Q/ \mathbb Z</math>, which is [[group isomorphism|isomorphic]] to the ''p''-[[quasicyclic group]] <math>\mathbb Z[p^\infty]</math>, is divisible.
* The multiplicative group of the [[complex number]]s <math>\mathbb C^*</math> is divisible.
* Every [[existentially closed]] abelian group (in the [[model theory|model theoretic]] sense) is divisible.

==Properties==
* If a divisible group is a [[subgroup]] of an abelian group then it is a [[direct summand]] of that abelian group.<ref>Hall, p.197</ref>
* Every abelian group can be [[Embedding|embedded]] in a divisible group.<ref>Griffith, p.17</ref> Put another way, the category of abelian groups [[Injective object#Enough injectives and injective hulls|has enough injectives]].
* Non-trivial divisible groups are not [[finitely generated abelian group|finitely generated]].
* Further, every abelian group can be embedded in a divisible group as an [[essential subgroup]] in a unique way.<ref>Griffith, p.19</ref>
* An abelian group is divisible if and only if it is ''p''-divisible for every prime ''p''.
* Let <math>A</math> be a [[Ring (mathematics)|ring]]. If <math>T</math> is a divisible group, then <math>\mathrm{Hom}_{\mathbf{Z}\text{-Mod}} (A,T)</math> is injective in the [[Category (mathematics)|category]] of <math>A</math>-[[Module (mathematics)|modules]].<ref>Lang, p. 106</ref>

==Structure theorem of divisible groups==

Let ''G'' be a divisible group. Then the [[torsion subgroup]] Tor(''G'') of ''G'' is divisible. Since a divisible group is an [[injective module]], Tor(''G'') is a [[direct summand]] of ''G''. So

:<math>G = \mathrm{Tor}(G) \oplus  G/\mathrm{Tor}(G).</math>

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

:<math>G/\mathrm{Tor}(G) = \bigoplus_{i \in I} \mathbb Q = \mathbb Q^{(I)}.</math>

The structure of the torsion subgroup is harder to determine, but one can show{{sfn|Kaplansky|1965}}{{sfn|Fuchs|1970}} that for all [[prime number]]s ''p'' there exists <math>I_p</math> such that

:<math>(\mathrm{Tor}(G))_p = \bigoplus_{i \in I_p} \mathbb Z[p^\infty] = \mathbb Z[p^\infty]^{(I_p)},</math>

where <math>(\mathrm{Tor}(G))_p</math> is the ''p''-primary component of Tor(''G'').

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

:<math>G = \left(\bigoplus_{p \in \mathbf P} \mathbb Z[p^\infty]^{(I_p)}\right) \oplus \mathbb Q^{(I)}.</math>

The cardinalities of the sets ''I'' and ''I''<sub>''p''</sub> for ''p''&nbsp;&isin;&nbsp;'''P''' are uniquely determined by the group ''G''.

==Injective envelope==
{{main|Injective envelope}}
As stated above, any abelian group ''A'' can be uniquely embedded in a divisible group ''D'' as an [[essential subgroup]].  This divisible group ''D'' is the '''injective envelope''' of ''A'', and this concept is the [[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.<ref>Griffith, p.7</ref> This is a special feature of [[hereditary ring]]s like the integers '''Z''': the [[direct sum of modules|direct sum]] of injective modules is injective because the ring is [[Noetherian ring|Noetherian]], 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 {{harv|Matlis|1958}}: 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]].

==Generalization==

Several distinct definitions generalize divisible groups to divisible modules. The following definitions have been used in the literature to define a '''divisible [[Module (mathematics)|module]]''' ''M'' over a [[Ring (mathematics)|ring]] ''R'':
# ''rM''&thinsp;=&thinsp;''M'' for all nonzero ''r'' in ''R''.{{sfn|Feigelstock|2006}} (It is sometimes required that ''r'' is not a zero-divisor, and some authors{{sfn|Cartan|Eilenberg|1999}} require that ''R'' is a [[Domain (ring theory)|domain]].)
# For every principal left [[Ideal (ring theory)|ideal]] ''Ra'', any [[Module homomorphism|homomorphism]] from ''Ra'' into ''M'' extends to a homomorphism from ''R'' into ''M''.{{sfn|Lam|1999}}{{sfn|Nicholson|Yousif
|2003}} (This type of divisible module is also called ''principally injective module''.)
# For every [[finitely generated module|finitely generated]] left ideal ''L'' of ''R'', any homomorphism from ''L'' into ''M'' extends to a homomorphism from ''R'' into ''M''.{{Citation needed|date=January 2023}}

The last two conditions are "restricted versions" of the [[Baer's criterion]] for [[injective module]]s. 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.{{sfn|Lam|1999|loc=p.70—73}} 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 ring|commutative]] domain, then the injective ''R'' modules coincide with the divisible ''R'' modules if and only if ''R'' is a [[Dedekind domain]].{{sfn|Lam|1999|loc=p.70—73}}

==See also==

* [[Injective object]]
* [[Injective module]]

==Notes==
{{reflist|colwidth=30em}}

==References==
*{{citation |last1=Cartan |first1=Henri  |author1-link=Henri Cartan|last2=Eilenberg |first2=Samuel |author2-link= Samuel Eilenberg|title=Homological algebra |series=Princeton Landmarks in Mathematics |publisher=Princeton University Press |place=Princeton, NJ |year=1999 |pages=xvi+390 |isbn=0-691-04991-2 |mr=1731415}} With an appendix by David A. Buchsbaum; Reprint of the 1956 original 
*{{citation |last=Feigelstock |first=Shalom |title=Divisible is injective |journal=Soochow J. Math. |volume=32 |year=2006 |number=2|pages=241–243 |issn=0250-3255 |mr=2238765}}
* {{cite book | last=Griffith|first=Phillip A. | title=Infinite Abelian group theory | series=Chicago Lectures in Mathematics | publisher=University of Chicago Press | year=1970 | isbn=0-226-30870-7 }}
* {{cite book | title=The theory of groups |last=Hall|first=Marshall Jr | authorlink=Marshall Hall (mathematician) | location=New York | publisher=Macmillan | year=1959 }}  Chapter 13.3.
* {{cite book | title=Infinite Abelian Groups|last=Kaplansky|first=Irving | authorlink=Irving Kaplansky | publisher=University of Michigan Press | year=1965 }}
* {{cite book | title=Infinite Abelian Groups Vol 1|last=Fuchs|first=László | authorlink=László Fuchs | publisher=Academic Press | year=1970 }}
*{{Citation | last1=Lam | first1=Tsit-Yuen | title=Lectures on modules and rings | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Graduate Texts in Mathematics No. 189 | isbn=978-0-387-98428-5 | mr=1653294 | year=1999 | volume=189 | doi=10.1007/978-1-4612-0525-8}}
* {{cite book | title=Algebra, Second Edition | author=Serge Lang | authorlink=Serge Lang | location=Menlo Park, California | publisher=Addison-Wesley | year=1984 }}
*{{Cite journal | last=Matlis|first=Eben | title=Injective modules over Noetherian rings | url=http://projecteuclid.org/getRecord?id=euclid.pjm/1103039896 | mr=0099360 | year=1958 | journal=Pacific Journal of Mathematics | issn=0030-8730 | volume=8 |issue=3 | pages=511–528 | doi=10.2140/pjm.1958.8.511| doi-access=free }}
*{{citation |last1=Nicholson|first1=W. K. |last2=Yousif|first2=M. F. |title=Quasi-Frobenius rings |series=Cambridge Tracts in Mathematics |volume=158 |publisher=Cambridge University Press |place=Cambridge |year=2003 |pages=xviii+307 |isbn=0-521-81593-2 |mr=2003785 |doi=10.1017/CBO9780511546525}}

[[Category:Abelian group theory]]
[[Category:Properties of groups]]