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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.
Some authors draw a distinction between an internal direct product and an external direct product. For example, if and are subgroups of an additive abelian group , such that + = and = {}, then , and we say that is the internal direct product of and .
A further sub-division into systems is defined by the rotational group G in the leftmost column then into rows of Laue classes. Every point group in a Laue class has exactly the same abstract group structure except the centred group in the rightmost column which is the direct product of the rotational group with inversion.
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.
These groups include the cyclic groups, the dicyclic groups, and the direct products of two cyclic groups. The polycyclic groups generalize metacyclic groups by allowing more than one level of group extension. A group is polycyclic if it has a finite descending sequence of subgroups, each of which is normal in the previous subgroup with a ...
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 ...
In mathematics, a product of groups usually refers to a direct product of groups, but may also mean: semidirect product; Product of group subsets;
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.