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  2. Direct sum of groups - Wikipedia

    en.wikipedia.org/wiki/Direct_sum_of_groups

    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 ...

  3. Direct sum - Wikipedia

    en.wikipedia.org/wiki/Direct_sum

    The direct sum is an operation between structures in abstract algebra, a branch of mathematics.It is defined differently but analogously for different kinds of structures. As an example, the direct sum of two abelian groups and is another abelian group consisting of the ordered pairs (,) where and

  4. Abelian group - Wikipedia

    en.wikipedia.org/wiki/Abelian_group

    The fundamental theorem of finite abelian groups states that every finite abelian group can be expressed as the direct sum of cyclic subgroups of prime-power order; it is also known as the basis theorem for finite abelian groups. Moreover, automorphism groups of cyclic groups are examples of abelian groups. [13]

  5. Direct product of groups - Wikipedia

    en.wikipedia.org/wiki/Direct_product_of_groups

    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 .

  6. Coproduct - Wikipedia

    en.wikipedia.org/wiki/Coproduct

    For example, the coproduct in the category of groups, called the free product, is quite complicated. On the other hand, in the category of abelian groups (and equally for vector spaces), the coproduct, called the direct sum, consists of the elements of the direct product which have only finitely many nonzero terms. (It therefore coincides ...

  7. Graded ring - Wikipedia

    en.wikipedia.org/wiki/Graded_ring

    In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups such that ⁠ + ⁠.The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid.

  8. Finitely generated abelian group - Wikipedia

    en.wikipedia.org/wiki/Finitely_generated_abelian...

    The primary decomposition formulation states that every finitely generated abelian group G is isomorphic to a direct sum of primary cyclic groups and infinite cyclic groups. A primary cyclic group is one whose order is a power of a prime. That is, every finitely generated abelian group is isomorphic to a group of the form

  9. Free abelian group - Wikipedia

    en.wikipedia.org/wiki/Free_abelian_group

    Every set can be the basis of a free abelian group, which is unique up to group isomorphisms. The free abelian group for a given basis set can be constructed in several different but equivalent ways: as a direct sum of copies of the integers, as a family of integer-valued functions, as a signed multiset, or by a presentation of a group.