Isomorphism to C_n with n prime

[Pengwuino]

Homework Statement

Prove taht if the order n of a group G is a prime number, then G must be isomorphic to the cyclic group fo order n, $$C_n$$.

The Attempt at a Solution

We have previously proven that a group can can be written as $$S = \{A,A^2,A^3,A^4...,A^n = E\}$$ where E is the identity and the group is of order n. We also have Lagrange that tells us, in this case, the order of every element in S is n if the order is prime.

So let's say we have the group $$G = \{A, A^2, A^3,..,A^g\}$$ where G is of order g which is prime and the cyclic group $$C_n = \{C, C^2, C^3,...,C^n\}$$ where n is again the prime order of the group. By this we know that $$A^m \ne E , m<g$$ and similarly $$C^m \ne E ,m<n$$.

Now it seems like you can make an absolutely arbitrary 1 to 1 mapping from $$G -> C_n$$, so is my best bet to try to prove that it's possible to make a non-1to1 mapping and show that it must not work?

 

[ystael]

How much data about a homomorphism $$\varphi: G \to C_n$$ do you need to completely determine $$\varphi$$? It isn't much.

 

[Pengwuino]

I suppose you just need to map one element and then by construction, the rest is figured out. That is, if I pick some $$A_1 \epsilon G$$ and map it to some $$C_1 \epsilon C_n$$, then I should be able to say $$A_1^2 = A_2 = C_1^2=C_2$$ which gives me a unique mapping I would think.