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."
is surjective,
and 
and 
are injective, then 
is injective;
is injective,
and 
and 
are surjective, then 
is surjective.
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.
, 
, 
,
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."
