divisible group

An abelian group $(G,\; +)$ is divisible if, for every positive integer $n$ and every $g\; \backslash in\; G$, there exists $y\; \backslash in\; G$ such that $ny=g$.Griffith, p.6 An equivalent condition is: for any positive integer $n$, $nG=G$, since the existence of $y$ for every $n$ and $g$ implies that $n\; G\backslash supseteq\; G$, and the other direction $n\; G\backslash subseteq\; G$ is true for every group. A third equivalent condition is that an abelian group $G$ is divisible if and only if $G$ 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 $p$-divisible for a prime $p$ if for every $g\; \backslash in\; G$, there exists $y\; \backslash in\; G$ such that $py=g$. Equivalently, an abelian group is $p$-divisible if and only if $pG=G$.

• The rational numbers $\backslash mathbb\; Q$ form a divisible group under addition.
• More generally, the underlying additive group of any vector space over $\backslash mathbb\; Q$ is divisible.
• Every quotient of a divisible group is divisible. Thus, $\backslash mathbb\; Q/\backslash mathbb\; Z$ is divisible.
• The p-primary component $\backslash mathbb\; Z[1/p]/\backslash mathbb\; Z$ of $\backslash mathbb\; Q/\; \backslash mathbb\; Z$, which is isomorphic to the p-quasicyclic group $\backslash mathbb\; Z[p^\backslash infty]$, is divisible.
• The multiplicative group of the complex numbers $\backslash mathbb\; C^*$ is divisible.
• Every existentially closed abelian group (in the model theoretic sense) is divisible.

• If a divisible group is a subgroup of an abelian group then it is a direct summand of that abelian group.Hall, p.197
• Every abelian group can be embedded in a divisible group.Griffith, p.17 Put another way, the category of abelian groups has enough injectives.
• Non-trivial divisible groups are not finitely generated.
• Further, every abelian group can be embedded in a divisible group as an essential subgroup in a unique way.Griffith, p.19
• An abelian group is divisible if and only if it is p-divisible for every prime p.
• Let $A$ be a ring. If $T$ is a divisible group, then $\backslash mathrm\{Hom\}\_\{\backslash mathbf\{Z\}\backslash text\{-Mod\}\}\; (A,T)$ is injective in the category of $A$-modules.Lang, p. 106

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

:$G\; =\; \backslash mathrm\{Tor\}(G)\; \backslash oplus\; G/\backslash mathrm\{Tor\}(G).$

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

:$G/\backslash mathrm\{Tor\}(G)\; =\; \backslash bigoplus\_\{i\; \backslash in\; I\}\; \backslash mathbb\; Q\; =\; \backslash mathbb\; Q^\{(I)\}.$

The structure of the torsion subgroup is harder to determine, but one can show{{sfn|Kaplansky|1965}}{{sfn|Fuchs|1970}} that for all prime numbers p there exists $I\_p$ such that

:$(\backslash mathrm\{Tor\}(G))\_p\; =\; \backslash bigoplus\_\{i\; \backslash in\; I\_p\}\; \backslash mathbb\; Z[p^\backslash infty]\; =\; \backslash mathbb\; Z[p^\backslash infty]^\{(I\_p)\},$

where $(\backslash mathrm\{Tor\}(G))\_p$ is the p-primary component of Tor(G).

Thus, if P is the set of prime numbers,

:$G\; =\; \backslash left(\backslash bigoplus\_\{p\; \backslash in\; \backslash mathbf\; P\}\; \backslash mathbb\; Z[p^\backslash infty]^\{(I\_p)\}\backslash right)\; \backslash oplus\; \backslash mathbb\; Q^\{(I)\}.$

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

{{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.

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.Griffith, p.7 This is a special feature of hereditary rings like the integers Z: the direct sum of injective modules is injective because the ring is 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.

Several distinct definitions generalize divisible groups to divisible modules. The following definitions have been used in the literature to define a divisible module M over a ring R:

1. rM = 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.)
2. For every principal left ideal Ra, any 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.)

1. For every 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 modules. 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 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}}

• Injective object
• Injective module

{{reflist|colwidth=30em}}

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Category:Abelian group theory

Category:Properties of groups