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The group operation in the external direct sum is pointwise multiplication, as in the usual direct product. This subset does indeed form a group, and for a finite set of groups {H i} the external direct sum is equal to the direct product. If G = ΣH i, then G is isomorphic to Σ E {H i}. Thus, in a sense, the direct sum is an "internal ...
General case: [2] In category theory the direct sum is often, but not always, the coproduct in the category of the mathematical objects in question. For example, in the category of abelian groups, direct sum is a coproduct. This is also true in the category of modules.
An abelian group is called torsion-free if every non-zero element has infinite order. Several classes of torsion-free abelian groups have been studied extensively: Free abelian groups, i.e. arbitrary direct sums of ; Cotorsion and algebraically compact torsion-free groups such as the -adic integers
In the context of abelian groups, the direct product is sometimes referred to as the direct sum, and is denoted . Direct sums play an important role in the classification of abelian groups: according to the fundamental theorem of finite abelian groups , every finite abelian group can be expressed as the direct sum of cyclic groups .
A free module is a module that can be represented as a direct sum over its base ring, so free abelian groups and free -modules are equivalent concepts: each free abelian group is (with the multiplication operation above) a free -module, and each free -module comes from a free abelian group in this way. [21]
Then, G/tG is a torsion-free abelian group and thus it is free abelian. tG is a direct summand of G, which means there exists a subgroup F of G s.t. =, where /. Then, F is also free abelian. Since tG is finitely generated and each element of tG has finite order, tG is finite.
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 ...
The two Prüfer theorems follow from a general criterion of decomposability of an abelian group into a direct sum of cyclic subgroups due to L. Ya. Kulikov: An abelian p -group A is isomorphic to a direct sum of cyclic groups if and only if it is a union of a sequence { A i } of subgroups with the property that the heights of all elements of A ...