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.
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 $\displaystyle f(a)f(b) = f(ab)$. So it does not matter if you multiply $\displaystyle a$ and $\displaystyle b$ 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 $\displaystyle |a| = \sqrt{a^2}$. The solution just sort of pops out this way.