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.(Crying)

Printable View

- Nov 20th 2011, 04:29 PMMAdoneProving isomorphism to Zn
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.(Crying)

- Nov 20th 2011, 04:35 PMJose27Re: Proving isomorphism to Zn
Let be a generator of your cyclic group , can you find a function such that and such that is an homomorphism?

- Nov 20th 2011, 04:46 PMMAdoneRe: Proving isomorphism to Zn
are you suggesting a concrete f like f(x)=x+5 or generalized like f(x)=ix+j.

- Nov 20th 2011, 04:49 PMTheChazRe: Proving isomorphism to Zn
Something concrete, in terms of things we know must exist (like generator "a").

Try a^n - Nov 20th 2011, 04:53 PMJose27Re: Proving isomorphism to Zn
- Nov 20th 2011, 05:39 PMMAdoneRe: Proving isomorphism to Zn
all elements of the cyclic group

- Nov 20th 2011, 09:04 PMDevenoRe: Proving isomorphism to Zn
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.