In group theory one defines the direct product of two
groups (*G*, *) and (*H*, o) , denoted by *G*×*H*, as follows:

- as set of the elements of the new group, take the
*cartesian product*of the sets of elements of*G*and*H*, that is {(*g*,*h*):*g*in*G*,*h*in*H*}; - on these elements put an operation, defined elementwise:
( *g*,*h*) × (*g'*,*h'*) = (*g***g'*,*h*o*h'*)

This construction gives a new group. It has a normal subgroup
isomorphic to *G* (given by the elements of the form (*g*, 1)),
and one isomorphic to *H* (comprising the elements (1, *h*)).

The reverse also holds, there is the following recognition theorem: If a group *K* contains two normal subgroups *G* and *H*, such that *K*= *GH* and the intersection of *G* and *H* contains only the identity, then *K* = *G* x *H*. A relaxation of these conditions gives the semidirect product.

As an example, take as *G* and *H* two copies of the unique (up to
isomorphisms) group of order 2, *C*_{2}: say {1, *a*} and {1, *b*}. Then *C*_{2}×*C*_{2} = {(1,1), (1,*b*), (*a*,1), (*a*,*b*)}, with the operation element by element. For instance, (1,*b*)*(*a*,1) = (1**a*, *b**1) = (*a*,*b*), and (1,*b*)*(1,*b*) = (1,*b*^{2}) = (1,1).

With a direct product, we get some natural group homomorphisms for free: the projection maps

- ,

Also, every homomorphism *f* on the direct product is totally determined by its component functions
.

The preceding two observations of the direct product makes the direct product the product in the category of groups. In a general category, given a collection of objects *A _{i}*

For groups we similarly define the direct product of a more general, arbitrary collection of groups *G _{i}* for

- ,

More abstract formulations and generalizations of the categorical product (as if this weren't abstract or general enough!) can be found in the separate entry on categorical products.

The product for vector spaces is very similar to the one defined for groups above, using the cartesian product with the operation of addition being componentwise, and the scalar multiplication just distributing over all the components (an easy generalization of how it is defined for **R**^{n}).

The direct product for a collection of topological spaces *X _{i}* for

The product topology for infinite products has a twist, and this has to do with being able to make all the projection maps continuous and to make all functions into the product continuous if and only if all its component functions are continuous (i.e. to satisfy the categorical definition of product: the morphisms here are continuous functions): we take as a basis of open sets to be the collection of all cartesian products of open subsets from each factor, as before, with the proviso that all but finitely many factors are the entire space:

Products (with the product topology) are nice with respect to preserving properties of their factors; for example, the product of Hausdorff spaces is Hausdorff; the product of connected spaces is connected, and the product of compact spaces is compact. That last one, called Tychonoff's theorem, is yet another equivalence to the axiom of choice.

For more properties and equivalent formulations, see the separate entry product topology.

*Still working on this... add stuff here if you like... thought it was going to be fast but have to make sure I understand the definitions just right...*

In general category theory, as described above in the groups section, a related and "dual" concept to the product is what is called the **categorical sum** or **coproduct**, which basically reverses all the arrows in the diagrams. In full detail, given a collection of objects *A _{j}*

Despite this innocuous-looking change in the name and notation, coproducts can be dramatically different from products. The coproduct in the category of sets is simply the disjoint union, the maps *i _{j}* being the inclusions. Unlike direct products, coproducts in other categories are not all obviously based on the notion for sets, because unions don't behave well with respect to preserving operations (e.g. the union of two groups need not be a group), and so coproducts in different categories can be dramatically different from each other. 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 (including vector spaces), the coproduct, called the

But despite all this dissmilarity, there is still, at the heart of the whole thing, a disjoint union: the direct sum of abelian groups is the group generated by the "almost" disjoint union (disjoint union of all nonzero elements, together with a common zero), similarly for vector spaces: the space spanned by the "almost" disjoint union; the free product for groups is generated by the set of all letters from a similar "almost disjoint" union where no two elements from different sets are allowed to commute.

See also: Free product, Cartesian product, Group, Direct sum.