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Isomorphism to C_n with n prime

  1. Feb 19, 2010 #1


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    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?
    Last edited: Feb 19, 2010
  2. jcsd
  3. Feb 19, 2010 #2
    How much data about a homomorphism [tex]\varphi: G \to C_n[/tex] do you need to completely determine [tex]\varphi[/tex]? It isn't much.
  4. Feb 21, 2010 #3


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