I have proven cyclic group to be a group, but now must prove cyclic group order n to be isomorphic to Zn. I realize I must show homomorphism, injection and surjection. But not struggling with initial equivalences. Please help.
MAdone, this is really the whole of the idea.
if is to be a homomorphism, we must have f(a*a) = f(a) + f(a) = 1+1. so there is really only "one" essential way to define f:
. that f is a homomorphism follows immediately from the laws of exponents.
since G is finite it suffices to show that f is surjective (clearly |G| = = n), and thus f is an isomorphism.