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In mathematics, specifically in group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H.This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.
The direct product for modules (not to be confused with the tensor product) is very similar to the one that is defined for groups above by using the Cartesian product with the operation of addition being componentwise, and the scalar multiplication just distributing over all the components.
The direct product of two groups N and H can be thought of as the semidirect product of N and H with respect to φ(h) = id N for all h in H. Note that in a direct product, the order of the factors is not important, since N × H is isomorphic to H × N. This is not the case for semidirect products, as the two factors play different roles.
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces.
This right group is the direct product of a group (positive real numbers under multiplication) and a right zero semigroup induced by the real numbers. Structurally, this is identical to formula 1 above. In fact, this is how all right group operations look like when written as ordered pairs of the direct product of their factors.
In mathematics, a product of groups usually refers to a direct product of groups, but may also mean: semidirect product; Product of group subsets; wreath product;
The direct product of groups consists of tuples of an element from each group in the product, with componentwise addition. The direct product of two free abelian groups is itself free abelian, with basis the disjoint union of the bases of the two groups. [8] More generally the direct product of any finite number of free abelian groups is free ...
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 ...