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Divisible group

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 nth multiple for each positive integer n. Divisible groups are important in understanding the structure of abelian groups, especially because…

Examples, Properties & Generalization

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Overview

Definition

Examples

Properties

Structure theorem of divisible groups

Injective envelope

Reduced abelian groups

Generalization

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Map overview Semantic statistics

Divisible group

Nodes48
Edges47
Triples34
Avg. degree1.96
Density0.041667
Components1

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Divisible group

Top relations

related to Properties · 9
Divisible group → An, Every, Further, Hom, If, Let, Mod, Non-trivial, Put
related to Structure theorem of divisible groups · 8
Divisible group → G/Tor, Let, Moreover, Since, So As, Then, Thus, Tor
related to Generalization · 6
Divisible group → For, It, Ra, Several, The, This
related to Examples · 4
Divisible group → Every, More, The, Thus
is a · 3
Divisible group → abelian group in which every element can, injective module, subgroup of an abelian group then it is a direct summand of that abelian group.Every abelian group can be embedded in a divisible group
related to Definition · 2
Divisible group → An, Equivalently
related to Injective envelope · 2
Divisible group → As, This

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Important terminology

divisible group abelian injective every displaystyle groups subgroup modules mathematics ring module domain mr category prime ideal isbn mathbb direct

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Divisible groupis aabelian group in which every element can0.90text
Divisible groupis asubgroup of an abelian group then it is a direct summand of that abelian group.Every abelian group can be embedded in a divisible group0.90text
Divisible groupis ainjective module0.90text
Divisible grouprelated to DefinitionAn0.60section
Divisible grouprelated to DefinitionEquivalently0.60section
Divisible grouprelated to ExamplesThe0.60section
Divisible grouprelated to ExamplesMore0.60section
Divisible grouprelated to ExamplesEvery0.60section
Divisible grouprelated to ExamplesThus0.60section
Divisible grouprelated to GeneralizationSeveral0.60section
Divisible grouprelated to GeneralizationThe0.60section
Divisible grouprelated to GeneralizationIt0.60section

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