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In the case of finitely generated abelian groups, this theorem guarantees that an abelian group splits as a direct sum of a torsion group and a free abelian group. The former may be written as a direct sum of finitely many groups of the form Z / p k Z {\displaystyle \mathbb {Z} /p^{k}\mathbb {Z} } for p {\displaystyle p} prime, and the latter ...
The fundamental theorem of finitely generated abelian groups can be stated two ways, generalizing the two forms of the fundamental theorem of finite abelian groups.The theorem, in both forms, in turn generalizes to the structure theorem for finitely generated modules over a principal ideal domain, which in turn admits further generalizations.
One important structure theorem of abelian varieties is Matsusaka's theorem. It states that over an algebraically closed field every abelian variety A {\displaystyle A} is the quotient of the Jacobian of some curve; that is, there is some surjection of abelian varieties J → A {\displaystyle J\to A} where J {\displaystyle J} is a Jacobian.
In mathematics, specifically in group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H.This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.
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The theorem states that every linearly ordered abelian group G can be embedded as an ordered subgroup of the additive group endowed with a lexicographical order, where is the additive group of real numbers (with its standard order), Ω is the set of Archimedean equivalence classes of G, and is the set of all functions from Ω to which vanish outside a well-ordered set.
The first Prüfer theorem states that an abelian group of bounded exponent is isomorphic to a direct sum of cyclic groups. The second Prüfer theorem states that a countable abelian p-group whose non-trivial elements have finite p-height is isomorphic to a direct sum of cyclic groups. Examples show that the assumption that the group be ...
An abelian group A is torsion-free if and only if it is flat as a Z-module, which means that whenever C is a subgroup of some abelian group B, then the natural map from the tensor product C ⊗ A to B ⊗ A is injective. Tensoring an abelian group A with Q (or any divisible group) kills torsion. That is, if T is a torsion group then T ⊗ Q = 0.