The intersection is a normal subgroup of , and The quotient groups and are isomorphic. Technically, it is not necessary for to be a normal subgroup, as long as is a subgroup of the normalizer of in . In this case, the intersection is not a normal subgroup of , but it is still a normal subgroup of . This theorem is sometimes called the isomorphism theorem, diamond theorem or the parallelogram theorem. An application of the second isomorphism theorem identifies projective linear groups: for example, the group on the complex projective line starts with setting , the group of invertible 2 × 2 complex matrices, , the subgroup of determinant 1 matrices, and the normal subgroup of scalar matrices , we have , where is the identity matrix, and . Then the second isomorphism theorem states that: Theorem C (groups) Let be a group, and a normal subgroup of . Then If is a subgroup of such that , then has a subgroup isomorphic to . Every subgroup of is of the form for some subgroup of such that . If is a normal subgroup of such that , then has a normal subgroup isomorphic to . Every normal subgroup of is of the form for some normal subgroup of such that . If is a normal subgroup of such that , then the quotient group is isomorphic to . Theorem D (groups) The correspondence theorem (also known as the lattice theorem) is sometimes called the third or fourth isomorphism theorem. The Zassenhaus lemma (also known as the butterfly lemma) is sometimes called the fourth isomorphism theorem. Discussion The first isomorphism theorem can be expressed in category theoretical language by saying that the category of groups is (normal epi, mono)-factorizable; in other words, the normal epimorphisms and the monomorphisms form a factorization system for the category. This is captured in the commutative diagram in the margin, which shows the objects and morphisms whose existence can be deduced from the morphism . The diagram shows that every morphism in the category of groups has a kernel in the category theoretical sense; the arbitrary morphism f factors into , where ι is a monomorphism and π is an epimorphism (in a conormal category, all epimorphisms are normal). This is represented in the diagram by an object and a monomorphism (kernels are always monomorphisms), which complete the short exact sequence running from