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To qualify as an abelian group, the set and operation, (,), must satisfy four requirements known as the abelian group axioms (some authors include in the axioms some properties that belong to the definition of an operation: namely that the operation is defined for any ordered pair of elements of A, that the result is well-defined, and that the ...
Every subgroup of a free abelian group is itself free abelian; this fact allows a general abelian group to be understood as a quotient of a free abelian group by "relations", or as a cokernel of an injective homomorphism between free abelian groups. The only free abelian groups that are free groups are the trivial group and the infinite cyclic ...
Every elementary abelian p-group is a vector space over the prime field with p elements, and conversely every such vector space is an elementary abelian group. By the classification of finitely generated abelian groups, or by the fact that every vector space has a basis, every finite elementary abelian group must be of the form (Z/pZ) n for n a ...
A group is an abelian group if and only if the derived group is trivial: [G,G] = {e}. Equivalently, if and only if the group equals its abelianization. See above for the definition of a group's abelianization. A group is a perfect group if and only if the derived group equals the group itself: [G,G] = G. Equivalently, if and only if the ...
An object in Ab is injective if and only if it is a divisible group; it is projective if and only if it is a free abelian group. The category has a projective generator (Z) and an injective cogenerator (Q/Z). Given two abelian groups A and B, their tensor product A⊗B is defined; it is again an abelian group.
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.
If a divisible group is a subgroup of an abelian group then it is a direct summand of that abelian group. [2] Every abelian group can be embedded in a divisible group. [3] 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 ...
The free abelian group on S can be explicitly identified as the free group F(S) modulo the subgroup generated by its commutators, [F(S), F(S)], i.e. its abelianisation. In other words, the free abelian group on S is the set of words that are distinguished only up to the order of letters. The rank of a free group can therefore also be defined as ...