divisible group

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

Some noteworthy facts:

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

单词	DivisibleGroup
释义	divisible group An abelian group $D$ is said to be divisible if for any $x\\in D$ , $n\\in\\mathbb{Z}^{+}$ , there exists an element $x^{\\prime}\\in D$ such that $nx^{\\prime}=x$ . Some noteworthy facts: • An abelian group is injective (http://planetmath.org/InjectiveModule) (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 $n$ copies of the group of rationals (for some cardinal number $n$ ).
随便看	WeakAIThesis weak AI thesis WeakApproximationTheorem weak approximation theorem WeakBisimulation weak bisimulation WeakConvergence weak convergence WeakConvergenceInNormedLinearSpace weak* convergence in normed linear space WeakDerivative weak derivative WeakDimensionOfAModule weak dimension of a module WeakerVersionOfStirlingsApproximation weaker version of Stirling’s approximation WeakestExtensionOfAPartialOrdering weakest extension of a partial ordering WeakGlobalDimension weak global dimension WeakHomotopyDoubleGroupoid weak homotopy double groupoid WeakHomotopyEquivalence weak homotopy equivalence WeakHopfAlgebra