Short five lemma

In mathematics, especially homological algebra and other applications of theory, the short five lemma is a special case of the five lemma. It states that for the following commutative diagram (in any abelian , or in the ), if the rows are short exact sequences, and if g and h are isomorphisms, then f is an isomorphism as well. It follows immediately from the five lemma. The essence of the lemma can be summarized as follows: if you have a homomorphism f from an object B to an object , and this homomorphism induces an isomorphism from a subobject A of B to a subobject of and also an isomorphism from the factor object B/A to /, then f itself is an isomorphism. Note however that the existence of f (such that the diagram commutes) has to be assumed from the start; two objects B and that simply have isomorphic sub- and factor objects need not themselves be isomorphic (for example, in the , B could be the cyclic group of order four and the Klein four-group).