is any group having subgroups and that satisfy the properties above, then is necessarily isomorphic to the direct product of and . In this situation, is sometimes referred to as the internal direct product of its subgroups and . In some contexts, the third property above is replaced by the following: 3′. Both and are normal in . This property is equivalent to property 3, since the elements of two normal subgroups with trivial intersection necessarily commute, a fact which can be deduced by considering the commutator of any in , in . Examples Presentations The algebraic structure of can be used to give a presentation for the direct product in terms of the presentations of and . Specifically, suppose that and where and are (disjoint) generating sets and and are defining relations. Then where is a set of relations specifying that each element of commutes with each element of . For example if and then Normal structure As mentioned above, the subgroups and are normal in . Specifically, define functions and by and . Then and are homomorphisms, known as projection homomorphisms, whose kernels are and , respectively. It follows that is an extension of by (or vice versa). In the case where is a finite group, it follows that the composition factors of are precisely the union of the composition factors of and the composition factors of . Further properties Universal property The direct product can be characterized by the following universal property. Let and be the projection homomorphisms. Then for any group and any homomorphisms and , there exists a unique homomorphism making the following diagram commute: Specifically, the homomorphism is given by the formula . This is a special case of the universal property for products in category theory. Subgroups If is a subgroup of and is a subgroup of , then the direct product is a subgroup of . For example, the isomorphic copy of in is the product , where is the trivial subgroup of . If and are normal, then is a normal subgroup of . Moreover, the quotient of the direct products is isomorphic to the direct product of the quotients: . Note that it is not true in general that every subgroup of is the product of a subgroup of with a subgroup of . For example, if is any non-trivial group, then the product has a diagonal subgroup which is