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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.
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 ...
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.
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.
The pushout of these maps is the direct sum of A and B. Generalizing to the case where f and g are arbitrary homomorphisms from a common domain Z, one obtains for the pushout a quotient group of the direct sum; namely, we mod out by the subgroup consisting of pairs (f(z), −g(z)). Thus we have "glued" along the images of Z under f and g.
The direct sum is a submodule of the direct product of the modules M i (Bourbaki 1989, §II.1.7). The direct product is the set of all functions α from I to the disjoint union of the modules M i with α(i)∈M i, but not necessarily vanishing for all but finitely many i. If the index set I is finite, then the direct sum and the direct product ...
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By the universal property of free groups, there exists a unique group homomorphism φ : F G → G whose restriction to G is the identity map. Let K be the kernel of this homomorphism. Then K is normal in F G , therefore is equal to its normal closure, so G | K = F G / K .