In mathematics, a direct product of objects already known can often be defined by giving a new one. That induces a structure on the Cartesian product of the underlying sets from that of the contributing objects. The categorical product is an abstraction of these notions in the setting of category theory.

Examples are the product of sets, groups (described below), rings, and other algebraic structures. The product of topological spaces is another instance.

The direct sum is a related operation that agrees with the direct product in some but not all cases.

In a similar manner, the direct product of finitely many algebraic structures can be talked about; for example, ${\displaystyle \mathbb {R} \times \mathbb {R} \times \mathbb {R} \times \mathbb {R} .}$ That relies on the direct product being associative up to isomorphism. That is, ${\displaystyle (A\times B)\times C\cong A\times (B\times C)}$ for any algebraic structures ${\displaystyle A,}$ ${\displaystyle B,}$ and ${\displaystyle C}$ of the same kind. The direct product is also commutative up to isomorphism; that is, ${\displaystyle A\times B\cong B\times A}$ for any algebraic structures ${\displaystyle A}$ and ${\displaystyle B}$ of the same kind. Even the direct product of infinitely many algebraic structures can be talked about; for example, the direct product of countably many copies of ${\displaystyle \mathbb {R} ,}$ is written as ${\displaystyle \mathbb {R} \times \mathbb {R} \times \mathbb {R} \times \dotsb .}$

In group theory, define the direct product of two groups ${\displaystyle (G,\circ )}$ and ${\displaystyle (H,\cdot ),}$ can be denoted by ${\displaystyle G\times H.}$ For abelian groups that are written additively, it may also be called the direct sum of two groups, denoted by ${\displaystyle G\oplus H.}$

It is defined as follows:

Note that ${\displaystyle (G,\circ )}$ may be the same as ${\displaystyle (H,\cdot ).}$

The construction gives a new group, which has a normal subgroup that is isomorphic to ${\displaystyle G}$ (given by the elements of the form ${\displaystyle (g,1)}$) and one that is isomorphic to ${\displaystyle H}$ (comprising the elements ${\displaystyle (1,h)}$).