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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 ...
A free abelian group with basis has the following universal property: for every function from to an abelian group , there exists a unique group homomorphism from to which extends . [ 4 ] [ 9 ] Here, a group homomorphism is a mapping from one group to the other that is consistent with the group product law: performing a product before or after ...
The Egyptians used the commutative property of multiplication to simplify computing products. [7] [8] Euclid is known to have assumed the commutative property of multiplication in his book Elements. [9] Formal uses of the commutative property arose in the late 18th and early 19th centuries, when mathematicians began to work on a theory of ...
The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian. In other words, / is abelian if and only if contains the commutator subgroup of . So in some sense it provides a measure of how far the group is from being abelian; the larger the ...
The Cayley table tells us whether a group is abelian. Because the group operation of an abelian group is commutative, a group is abelian if and only if its Cayley table's values are symmetric along its diagonal axis. The group {1, −1} above and the cyclic group of order 3 under ordinary multiplication are both examples of abelian groups, and ...
For a commutative ring R and R-modules A and B, Ext i R (A, B) is an R-module (using that Hom R (A, B) is an R-module in this case). For a non-commutative ring R, Ext i R (A, B) is only an abelian group, in general. If R is an algebra over a ring S (which means in particular that S is commutative), then Ext i R (A, B) is at least an S-module.
Moreover, for a fixed ring R, Tor is a functor in each variable (from R-modules to abelian groups). For a commutative ring R and R-modules A and B, Tor R i (A, B) is an R-module (using that A ⊗ R B is an R-module in this case). For a non-commutative ring R, Tor R i (A, B) is only an abelian group, in general.
Given a commutative monoid M, "the most general" abelian group K that arises from M is to be constructed by introducing inverse elements to all elements of M. Such an abelian group K always exists; it is called the Grothendieck group of M. It is characterized by a certain universal property and can also be concretely constructed from M.