same properties as the direct product of two groups, and can be characterized algebraically in a similar way. Infinite direct products It is also possible to take the direct product of an infinite number of groups. For an infinite sequence of groups, this can be defined just like the finite direct product of above, with elements of the infinite direct product being infinite tuples. More generally, given an indexed family { } of groups, the direct product is defined as follows: Unlike a finite direct product, the infinite direct product is not generated by the elements of the isomorphic subgroups { }. Instead, these subgroups generate a subgroup of the direct product known as the infinite direct sum, which consists of all elements that have only finitely many non-identity components. Other products Semidirect products Recall that a group with subgroups and is isomorphic to the direct product of and as long as it satisfies the following three conditions: The intersection is trivial. Every element of can be expressed uniquely as the product of an element of and an element of . Both and are normal in . A semidirect product of and is obtained by relaxing the third condition, so that only one of the two subgroups is required to be normal. The resulting product still consists of ordered pairs , but with a slightly more complicated rule for multiplication. It is also possible to relax the third condition entirely, requiring neither of the two subgroups to be normal. In this case, the group is referred to as a Zappa–Szép product of and . Free products The free product of and , usually denoted , is similar to the direct product, except that the subgroups and of are not required to commute. That is, if = | and = |, are presentations for and , then = |. Unlike the direct product, elements of the free product cannot be represented by ordered pairs. In fact, the free product of any two nontrivial groups is infinite. The free product is actually the coproduct in the category of groups. Subdirect products If and are groups, a subdirect product of and is any subgroup of which maps surjectively onto and under the projection homomorphisms. By Goursat's lemma, every subdirect product is a fiber product. Fiber products Let , , and be groups, and let and be homomorphisms. The fiber product of and over