Five Lemma
A diagram lemma which states that, given the commutative diagram of additive Abelian groups with exact rows, the following holds:
1. If 
is surjective, and 
and 
are injective, then 
is injective;
2. If 
is injective, and 
and 
are surjective, then 
is surjective.
If 
and 
are bijective, the hypotheses of (1) and (2) are satisfied simultaneously, and the conclusion is that 
is bijective. This statement is known as the Steenrod five lemma.
If 
, 
, 
, and 
are the zero group, then 
and 
are zero maps, and thus are trivially injective and surjective. In this particular case the diagram reduces to that shown above. It follows from (1), respectively (2), that 
is injective (or surjective) if 
and 
are. This weaker statement is sometimes referred to as the "short five lemma."
REFERENCES:
Eilenberg, S. and Steenrod, N. Foundations of Algebraic Topology. Princeton, NJ: Princeton University Press, p. 16, 1952.
Fulton, W. Algebraic Topology: A First Course. New York: Springer-Verlag, pp. 346-347, 1995.
Lang, S. Algebra, rev. 3rd ed. New York: Springer Verlag, p. 169, 2002.
Mac Lane, S. Homology. Berlin: Springer-Verlag, p. 14, 1967.
Mitchell, B. Theory of Categories. New York: Academic Press, pp. 35-36, 1965.
Munkres, J. R. Elements of Algebraic Topology. New York: Perseus Books Pub., p. 140, 1993.
Rotman, J. J. An Introduction to Algebraic Topology. New York: Springer-Verlag, pp. 98-99, 1988.
Spanier, E. H. Algebraic Topology. New York: McGraw-Hill, pp. 185-186, 1966.
