PlanetMath: divisible group

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divisible group

(Definition)

An abelian group is said to be divisible if for any $x\in D$ , $n\in\mathbb{Z}^+$ , there exists an element $x'\in D$ such that .

Some noteworthy facts:

An abelian group is injective (as a $\mathbb{Z}$ -module) if and only if it is divisible.
Every group is isomorphic to a subgroup of a divisible group.
Any divisible abelian group is isomorphic to the direct sum of its torsion subgroup and copies of the group of rationals (for some cardinal number ).

"divisible group" is owned by mathcam.

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Attachments:: example of divisible group (Example) by mathcam
-divisible group (Definition) by CWoo

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Cross-references: cardinal number, rationals, torsion subgroup, direct sum, subgroup, isomorphic, group, divisible, abelian group
There are 6 references to this entry.

This is version 4 of divisible group, born on 2003-07-23, modified 2003-07-24.
Object id is 4499, canonical name is DivisibleGroup.
Accessed 2803 times total.

Classification:

20K99 (Group theory and generalizations :: Abelian groups :: Miscellaneous)

Pending Errata and Addenda

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