The first group isomorphism theorem, also known as the fundamental homomorphism theorem, states that if 
is a group homomorphism,
then 
and 
, where 
indicates that 
is a normal subgroup of
, 
denotes the group kernel,
and 
indicates that 
and 
are isomorphic groups.
A corollary states that if 
is a group homomorphism,
then
1. 
is injective iff 
2. 
, where 
denotes the group order of
a group 
.
