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  2. Direct product - Wikipedia

    en.wikipedia.org/wiki/Direct_product

    The direct sum and direct product are not isomorphic for infinite indices, where the elements of a direct sum are zero for all but for a finite number of entries. They are dual in the sense of category theory: the direct sum is the coproduct, while the direct product is the product.

  3. Direct product of groups - Wikipedia

    en.wikipedia.org/wiki/Direct_product_of_groups

    Moreover, the quotient of the direct products is isomorphic to the direct product of the quotients: (G × H) / (A × B) ≅ (G / A) × (H / B). Note that it is not true in general that every subgroup of G × H is the product of a subgroup of G with a subgroup of H. For example, if G is any non-trivial group, then the product G × G has a ...

  4. Direct sum - Wikipedia

    en.wikipedia.org/wiki/Direct_sum

    The direct sum is also commutative up to isomorphism, i.e. for any algebraic structures and of the same kind. The direct sum of finitely many abelian groups, vector spaces, or modules is canonically isomorphic to the corresponding direct product. This is false, however, for some algebraic objects, like nonabelian groups.

  5. 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 ...

  6. Direct sum of modules - Wikipedia

    en.wikipedia.org/wiki/Direct_sum_of_modules

    In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with no "unnecessary" constraints, making it an example of a coproduct. Contrast with the direct product, which is the dual notion.

  7. Universal property - Wikipedia

    en.wikipedia.org/wiki/Universal_property

    Other objects that can be defined by universal properties include: all free objects, direct products and direct sums, free groups, free lattices, Grothendieck group, completion of a metric space, completion of a ring, Dedekind–MacNeille completion, product topologies, Stone–Čech compactification, tensor products, inverse limit and direct ...

  8. Kronecker product - Wikipedia

    en.wikipedia.org/wiki/Kronecker_product

    In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix.It is a specialization of the tensor product (which is denoted by the same symbol) from vectors to matrices and gives the matrix of the tensor product linear map with respect to a standard choice of basis.

  9. Free abelian group - Wikipedia

    en.wikipedia.org/wiki/Free_abelian_group

    The direct sum of infinitely many free abelian groups remains free abelian. It has a basis consisting of tuples in which all but one element is the identity, with the remaining element part of a basis for its group. [8] Every free abelian group may be described as a direct sum of copies of , with one copy for each member of its basis. [13] [14 ...