Originally Posted by
ChrisBickle OK this question was on a hw assignment i had and noone in the class could answer it and the teacher wouldnt give us the answer so its been bugging me. I think it is kinda like Lagrange's Theorem based on the conclusion but have no idea how to prove it i think the notation and inverses are screwing me up.
Let phi: G---->G' be a homomorphism of groups. Suppose a' = phi(a) element of Range(phi) Let f: ker(phi)--->G be defined as f(k) = ak. Show that f is a 1-1 onto correspondence between ker(phi) and phi(inverse)({a')}. Thus every element of phi(G) has the same sinze inverse image implying that |phi(G)||ker(phi)|=|G|
The only hint he gave was to show a 1-1 for f and for onto to show f(ker(phi)) = phi(inverse){a'}
I really dont have any work to show because i am so confused like i said all the inverses and stuff are throwing me off. I know everything in the kernel is mapped to e' and all a' go back to phi(inverse) of a' but that is all i got. Any help is greatly appreciated