Divisible group

<h2 id="definition">Definition</h2>
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}
 
.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> 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 <a href="/facts/Injective_object/JGABzfyw">injective object</a> in the <a href="/facts/Category_of_abelian_groups/jF3sXRFn">category of abelian groups</a>; for this reason, a divisible group is sometimes called an injective group.
An abelian group is 
 
 
 
 p
 
 
 {\displaystyle p}
 
-divisible for a <a href="/facts/Prime_number/L20FLkgn">prime</a> 
 
 
 
 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}
 
.

<h2 id="examples">Examples</h2>
<ul><li>The <a href="/facts/Rational_number/VJQmyY05">rational numbers</a> 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
 form a divisible group under addition.</li>
<li>More generally, the underlying additive group of any <a href="/facts/Vector_space/M1pxMLx2">vector space</a> over 
 
 
 
 
 Q
 
 
 
 {\displaystyle \mathbb {Q} }
 
 is divisible.</li>
<li>Every <a href="/facts/Quotient_group/HlwpHtjb">quotient</a> of a divisible group is divisible. Thus, 
 
 
 
 
 Q
 
 
 /
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Q} /\mathbb {Z} }
 
 is divisible.</li>
<li>The p-<a href="/facts/Primary_component/dgPs3BYw">primary component</a> 
 
 
 
 
 Z
 
 [
 1
 
 /
 
 p
 ]
 
 /
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Z} [1/p]/\mathbb {Z} }
 
 of 
 
 
 
 
 Q
 
 
 /
 
 
 Z
 
 
 
 {\displaystyle \mathbb {Q} /\mathbb {Z} }
 
, which is <a href="/facts/Group_isomorphism/LC4N1aDw">isomorphic</a> to the p-<a href="/facts/Quasicyclic_group/QtzBWXFv">quasicyclic group</a> 
 
 
 
 
 Z
 
 [
 
 p
 
 ∞
 
 
 ]
 
 
 {\displaystyle \mathbb {Z} [p^{\infty }]}
 
, is divisible.</li>
<li>The multiplicative group of the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> 
 
 
 
 
 
 C
 
 
 ∗
 
 
 
 
 {\displaystyle \mathbb {C} ^{*}}
 
 is divisible.</li>
<li>Every <a href="/facts/Existentially_closed/2vS2zPzh">existentially closed</a> abelian group (in the <a href="/facts/Model_theory/vLWVV8sQ">model theoretic</a> sense) is divisible.</li></ul>
<h2 id="properties">Properties</h2>
<ul><li>If a divisible group is a <a href="/facts/Subgroup/r91mBgpa">subgroup</a> of an abelian group then it is a <a href="/facts/Direct_summand/x3ljVsP6">direct summand</a> of that abelian group.<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a></li>
<li>Every abelian group can be <a href="/facts/Embedding/HKeVKc42">embedded</a> in a divisible group.<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> Put another way, the category of abelian groups <a href="/facts/Injective_object/JGABzfyw">has enough injectives</a>.</li>
<li>Non-trivial divisible groups are not <a href="/facts/Finitely_generated_abelian_group/dwtl7mp4">finitely generated</a>.</li>
<li>Further, every abelian group can be embedded in a divisible group as an <a href="/facts/Essential_subgroup/PWkXXhF7">essential subgroup</a> in a unique way.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a></li>
<li>An abelian group is divisible if and only if it is p-divisible for every prime p.</li>
<li>Let 
 
 
 
 A
 
 
 {\displaystyle A}
 
 be a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a>. 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 <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> of 
 
 
 
 A
 
 
 {\displaystyle A}
 
-<a href="/facts/Module_(mathematics)/jP0LIhFL">modules</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a></li></ul>
<h2 id="structure-theorem-of-divisible-groups">Structure theorem of divisible groups</h2>
Let G be a divisible group. Then the <a href="/facts/Torsion_subgroup/dIwLuG5j">torsion subgroup</a> Tor(G) of G is divisible. Since a divisible group is an <a href="/facts/Injective_module/1sAYFnyJ">injective module</a>, Tor(G) is a <a href="/facts/Direct_summand/x3ljVsP6">direct summand</a> 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 <a href="/facts/Torsion_(algebra)/uNRnwtLz">torsion-free</a>. 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<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a><a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> that for all <a href="/facts/Prime_number/L20FLkgn">prime numbers</a> 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 Ip for p ∈ P are uniquely determined by the group G.

<h2 id="injective-envelope">Injective envelope</h2>
Main article: <a href="/facts/Injective_envelope/VoENIwTw">Injective envelope</a>
As stated above, any abelian group A can be uniquely embedded in a divisible group D as an <a href="/facts/Essential_subgroup/PWkXXhF7">essential subgroup</a>. This divisible group D is the injective envelope of A, and this concept is the <a href="/facts/Injective_hull/VoENIwTw">injective hull</a> in the category of abelian groups.

<h2 id="reduced-abelian-groups">Reduced abelian groups</h2>
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.<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> This is a special feature of <a href="/facts/Hereditary_ring/w2yUaqFe">hereditary rings</a> like the integers Z: the <a href="/facts/Direct_sum_of_modules/Y6PBJnjg">direct sum</a> of injective modules is injective because the ring is <a href="/facts/Noetherian_ring/kYbRSdnp">Noetherian</a>, 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): 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 <a href="/facts/Ulm%2527s_theorem/mBkah65b">Ulm's theorem</a>.

<h2 id="generalization">Generalization</h2>
Several distinct definitions generalize divisible groups to divisible modules. The following definitions have been used in the literature to define a divisible <a href="/facts/Module_(mathematics)/jP0LIhFL">module</a> M over a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a> R:

<ol><li>rM = M for all nonzero r in R.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> (It is sometimes required that r is not a zero-divisor, and some authors<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> require that R is a <a href="/facts/Domain_(ring_theory)/p4eKRC8W">domain</a>.)</li>
<li>For every principal left <a href="/facts/Ideal_(ring_theory)/vCvytduX">ideal</a> Ra, any <a href="/facts/Module_homomorphism/SCSzvgxx">homomorphism</a> from Ra into M extends to a homomorphism from R into M.<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a><a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> (This type of divisible module is also called principally injective module.)</li>
<li>For every <a href="/facts/Finitely_generated_module/6Njcpgzr">finitely generated</a> left ideal L of R, any homomorphism from L into M extends to a homomorphism from R into M.</li></ol>
The last two conditions are "restricted versions" of the <a href="/facts/Baer%2527s_criterion/1sAYFnyJ">Baer's criterion</a> for <a href="/facts/Injective_module/1sAYFnyJ">injective modules</a>. 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.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> 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 <a href="/facts/Commutative_ring/QS1r2ovR">commutative</a> domain, then the injective R modules coincide with the divisible R modules if and only if R is a <a href="/facts/Dedekind_domain/tAsvAcaW">Dedekind domain</a>.<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Injective_object/JGABzfyw">Injective object</a></li>
<li><a href="/facts/Injective_module/1sAYFnyJ">Injective module</a></li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/Henri_Cartan/IGrXDS4a">Cartan, Henri</a>; <a href="/facts/Samuel_Eilenberg/vs6aksKX">Eilenberg, Samuel</a> (1999), Homological algebra, Princeton Landmarks in Mathematics, Princeton, NJ: Princeton University Press, pp. xvi+390, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-691-04991-2, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1731415">1731415</a> With an appendix by David A. Buchsbaum; Reprint of the 1956 original</li>
<li>Feigelstock, Shalom (2006), "Divisible is injective", Soochow J. Math., 32 (2): 241–243, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0250-3255">0250-3255</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2238765">2238765</a></li>
<li>Griffith, Phillip A. (1970). Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-226-30870-7.</li>
<li><a href="/facts/Marshall_Hall_(mathematician)/wLpjsYnL">Hall, Marshall Jr</a> (1959). The theory of groups. New York: Macmillan. Chapter 13.3.</li>
<li><a href="/facts/Irving_Kaplansky/nRLyp7K9">Kaplansky, Irving</a> (1965). Infinite Abelian Groups. University of Michigan Press.</li>
<li><a href="/facts/L%25C3%25A1szl%25C3%25B3_Fuchs/vNJh7RxD">Fuchs, László</a> (1970). Infinite Abelian Groups Vol 1. Academic Press.</li>
<li>Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, vol. 189, Berlin, New York: <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2F978-1-4612-0525-8">10.1007/978-1-4612-0525-8</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-98428-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1653294">1653294</a></li>
<li><a href="/facts/Serge_Lang/I7MW2rUu">Serge Lang</a> (1984). Algebra, Second Edition. Menlo Park, California: Addison-Wesley.</li>
<li>Matlis, Eben (1958). <a href="http://projecteuclid.org/getRecord?id=euclid.pjm/1103039896">"Injective modules over Noetherian rings"</a>. Pacific Journal of Mathematics. 8 (3): 511–528. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2140%2Fpjm.1958.8.511">10.2140/pjm.1958.8.511</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0030-8730">0030-8730</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0099360">0099360</a>.</li>
<li>Nicholson, W. K.; Yousif, M. F. (2003), Quasi-Frobenius rings, Cambridge Tracts in Mathematics, vol. 158, Cambridge: Cambridge University Press, pp. xviii+307, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2FCBO9780511546525">10.1017/CBO9780511546525</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-81593-2, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2003785">2003785</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Griffith, p.6 <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Hall, p.197 <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Griffith, p.17 <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Griffith, p.19 <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Lang, p. 106 <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Kaplansky 1965. - Kaplansky, Irving (1965). Infinite Abelian Groups. University of Michigan Press. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Fuchs 1970. - Fuchs, László (1970). Infinite Abelian Groups Vol 1. Academic Press. <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Griffith, p.7 <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Feigelstock 2006. - Feigelstock, Shalom (2006), "Divisible is injective", Soochow J. Math., 32 (2): 241–243, ISSN 0250-3255, MR 2238765 <a href="https://search.worldcat.org/issn/0250-3255" target="_blank">https://search.worldcat.org/issn/0250-3255</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Cartan & Eilenberg 1999. - Cartan, Henri; Eilenberg, Samuel (1999), Homological algebra, Princeton Landmarks in Mathematics, Princeton, NJ: Princeton University Press, pp. xvi+390, ISBN 0-691-04991-2, MR 1731415 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1731415" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1731415</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Lam 1999. - Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, vol. 189, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-0525-8, ISBN 978-0-387-98428-5, MR 1653294 <a href="https://doi.org/10.1007%2F978-1-4612-0525-8" target="_blank">https://doi.org/10.1007%2F978-1-4612-0525-8</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Nicholson & Yousif 2003. - Nicholson, W. K.; Yousif, M. F. (2003), Quasi-Frobenius rings, Cambridge Tracts in Mathematics, vol. 158, Cambridge: Cambridge University Press, pp. xviii+307, doi:10.1017/CBO9780511546525, ISBN 0-521-81593-2, MR 2003785 <a href="https://doi.org/10.1017%2FCBO9780511546525" target="_blank">https://doi.org/10.1017%2FCBO9780511546525</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Lam 1999, p.70—73. - Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, vol. 189, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-0525-8, ISBN 978-0-387-98428-5, MR 1653294 <a href="https://doi.org/10.1007%2F978-1-4612-0525-8" target="_blank">https://doi.org/10.1007%2F978-1-4612-0525-8</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Lam 1999, p.70—73. - Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, vol. 189, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-0525-8, ISBN 978-0-387-98428-5, MR 1653294 <a href="https://doi.org/10.1007%2F978-1-4612-0525-8" target="_blank">https://doi.org/10.1007%2F978-1-4612-0525-8</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
</ol>

Divisible group open-in-new

Divisible group