(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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, [tex]C_n[/tex].

3. The attempt at a solution

We have previously proven that a group can can be written as [tex]S = \{A,A^2,A^3,A^4...,A^n = E\}[/tex] 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 lets say we have the group [tex] G = \{A, A^2, A^3,..,A^g\}[/tex] where G is of order g which is prime and the cyclic group [tex] C_n = \{C, C^2, C^3,...,C^n\}[/tex] where n is again the prime order of the group. By this we know that [tex]A^m \ne E , m<g[/tex] and similarly [tex]C^m \ne E ,m<n[/tex].

Now it seems like you can make an absolutely arbitrary 1 to 1 mapping from [tex] G -> C_n[/tex], 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?

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# Homework Help: Isomorphism to C_n with n prime

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