They are! Consider the identity map from Z to Q. They are not isomorphic, but there is a homomorphism between them.

if x>3 then

This all depends on the type of field you are referring to. Homomorphism in general is math-structural map between two alike structures, such that:

g(xy) = g(x)g(y)

The special linear group, SL(n,R), is a normal subgroup of the general linear subgroup GL(n,R).

Proof: SL(n,R) is the kernel of the determinant function, which is a group homomorphism. The kernel of a group homomorphism is always a normal subgroup.

not really. The best example is the difference between two matemathical terms: "isomporphism" and "homomorphism"