The mapping f from R* to R* (R*=set of real numbers with no zero), defined by f(x)=|x|, is a homomorphism with Ker f = {1,-1}. How would I prove that this mapping is a homomorphism? I am really struggling with this concept, any help would be great.
It is multiplication on a group. There is no zero, and so it is not a ring and it is not addition.
A homomorphism is a function which preserves your operation (here, it is multiplication). That is, they are functions such that . So it does not matter if you multiply and together then apply the function, or if you multiply their images, you will always get the same element. These are nice, and they allow us to compare groups in a natural way.
I would attack this problem by using that fact that . The solution just sort of pops out this way.