Editing Short five lemma

{{Short description|Special case of the five lemma}}
In [[mathematics]], especially [[homological algebra]] and other applications of [[abelian category]] theory, the '''short five lemma''' is a special case of the [[five lemma]].
It states that for the following [[commutative diagram]] (in any abelian [[category (mathematics)|category]], or in the [[category of groups]]), if the rows are [[exact sequence|short exact sequences]], and if ''g'' and ''h'' are [[isomorphism]]s, then ''f'' is an isomorphism as well.

: [[Image:Short_5_lemma.svg]]

It follows immediately from the [[five lemma]].

The essence of the [[lemma (mathematics)|lemma]] can be summarized as follows: if you have a [[homomorphism]] ''f'' from an object ''B'' to an object ''{{prime|B}}'', and this homomorphism induces an isomorphism from a subobject ''A'' of ''B'' to a subobject ''{{prime|A}}'' of ''{{prime|B}}'' and also an isomorphism from the factor object ''B''/''A'' to ''{{prime|B}}''/''{{prime|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 ''{{prime|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]] of [[order of a group|order]] four and ''{{prime|B}}'' the [[Klein four-group]]).

== References ==
*{{cite book |first=Thomas W. |last=Hungerford |author1-link=Thomas W. Hungerford |title=Algebra |publisher=[[Springer-Verlag]] |location=Berlin |year=2003 | orig-date=1980 |page=176 |isbn=0-387-90518-9 | series=[[Graduate Texts in Mathematics]] | volume=73 | zbl=0442.00002 }}
* {{cite book | editor1-last=Pedicchio | editor1-first=Maria Cristina | editor2-last=Tholen | editor2-first=Walter | title=Categorical foundations. Special topics in order, topology, algebra, and sheaf theory | series=Encyclopedia of Mathematics and Its Applications | volume=97 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2004 | isbn=0-521-83414-7 | zbl=1034.18001 }}

{{DEFAULTSORT:Short Five Lemma}}
[[Category:Homological algebra]]
[[Category:Lemmas in category theory]]