Statements
Consider the following commutative diagram in any abelian category (such as the category of abelian groups or the category of vector spaces over a given field) or in the category of groups.
The five lemma states that, if the rows are exact, m and p are isomorphisms, l is an epimorphism, and q is a monomorphism, then n is also an isomorphism.
The two four-lemmas state:
(1) If the rows in the commutative diagram
are exact and m and p are epimorphisms and q is a monomorphism, then n is an epimorphism.
(2) If the rows in the commutative diagram
are exact and m and p are monomorphisms and l is an epimorphism, then n is a monomorphism.
Read more about this topic: Five Lemma
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