Homo Morphism

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.


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