Examples
Sgn
Det
Mod
Inclusion
The sign map sgn form Sn to {+1,-1}
is a group morphism. Its kernel is the subgroup An
of Sn.
The determinant is a morphism from the general linear group
GLn(R) to the multiplicative group
R*.
Its kernel is the special linear group
SLn(R).
The map from Z to Z/nZ defined by
x -> x mod n
is a morphism of the additive group on Z onto
Z/nZ.
The kernel is nZ.
If
H is a subgroup of a group
G, then the map
h
H ->
h
G
is a morphism.
Its kernel is trivial, i.e., consists of the identity only.