Mapping
Homomorphism mapping
Consider group
and mapping 
.For
, 
Isomorphism
If
is a bijective mapping,
Kernel
If
is the unit element of 
,
Homomorphism theorem
Consider homomorphism mapping
,
's kernel,
then ,
is an isomorphism mapping, and
Automorphism mapping
Automorphism mapping from G to G
Automorphism group
An identity mapping is an automorphism mapping and a composite automorphism mapping
is also an automorphism mapping. Moreover, the set of automorphism mappings forms a group.
Conjugate
If
exists and satisfies
and 
, then
are conjugates of each other.[0]Top