A homomorphism is a function from one
AlgebraicGroup into another which preserves the group structure. Formally, if
G and
H are groups with operations * and #, respectively, and
f is a function from
G to
H, then
f is a homomorphism if
-
- a * b = c implies f(a) # f(b) = f(c)
for all
a,
b, and
c in
G.
A homomorphism maps the identity element of
G onto the identity element of
H. If
h is the image of
g under a homomorphism, then the inverse of
h is the image of the inverse of
g.
The definition above is not the usual one, but it is equivalent to it. The usual definition is that
f(
a*
b) =
f(
a) #
f(
b) for all
a and
b in
G.
If a homomorphism is one-to-one and onto, then its inverse is also a homomorphism, and it is called an
isomorphism. Two groups are considered to be the same if there is an isomorphism between them.
If
H is a group with identity element
e, and
f is a homomorphism from
G to
H, then the set of all elements
g in
G such that
f(
g) =
e is called the
kernel of the homomorphism. The kernel of a homomorphism is always a
NormalSubgroup of
G.
CategoryMath