short five lemma | Steenrod five lemma

A diagram lemma which states that, given the commutative diagram of additive Abelian groups with exact rows, the following holds: 1. If f_0 is surjective, and f_1 and f_3 are injective, then f_2 is injective; 2. If f_4 is injective, and f_1 and f_3 are surjective, then f_2 is surjective. If f_0, f_1, f_3 and f_4 are bijective, the hypotheses of (1) and (2) are satisfied simultaneously, and the conclusion is that f_2 is bijective. This statement is known as the Steenrod five lemma.

commutative diagram | diagram lemma | exact sequence | four lemma