In mathematics, especially homological algebra and other applications of

abelian category In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...

theory, the short five lemma is a special case of the

five lemma In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams. The five lemma is not only valid for abelian categories but also work ...

. It states that for the following commutative diagram (in any abelian category, or in the category of

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

s), if the rows are short exact sequences, and if ''g'' and ''h'' are

isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...

s, then ''f'' is an isomorphism as well. It follows immediately from the

. The essence of the lemma can be summarized as follows: if you have a homomorphism ''f'' from an object ''B'' to an object ''B′'', and this homomorphism induces an isomorphism from a subobject ''A'' of ''B'' to a subobject ''A′'' of ''B′'' and also an isomorphism from the factor object ''B''/''A'' to ''B′''/''A′'', 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 ''B′'' that simply have isomorphic sub- and factor objects need not themselves be isomorphic (for example, in the category of abelian groups, ''B'' could be the

cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...

of order four and ''B′'' the

Klein four-group In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one. ...

References

* * {{DEFAULTSORT:Short Five Lemma Homological algebra Lemmas in category theory