Isomorphisms between cyclic groups!

  • #1
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Ok, here is something i thought i understood, but it turns out i am having difficulties fully grasping/proving it.

Let [tex]\theta:G->G'[/tex] be an isomorphism between G and G', where o(G)=m=o(G'), and both G and G' are cyclic, i.e. G=[a] and G'=

So my question is, when we want to find the total number of isomorphisms from G to G', i 'know' that the total number of such isomorphisms is basically
the number of generators of G':
[tex]\theta(a)=b^k[/tex] where gcd(k,m)=1. But i don't really understand why? That is, how are we sure that by just counting the number of generators on G' we are actually finding the total number of such isomorphisms. ?

Any explanations would be appreciated.
 

Answers and Replies

  • #2
HallsofIvy
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Suppose [itex]\theta[/itex] were any isomorphism from G to G'. Apply [itex]\theta[/itex] to a generator of G. It is simple to prove that [itex]\theta[/itex] MUST map a generator to a generator. So it is easy to see that it is a matter of how many generators each has: each mapping of generator to generator gives an isomorphism.
 
  • #3
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well, yeah, i know how to prove that an isomorphism perserves the order of each element, and thus must mapp a generator to a generator, but say for example that the generators of G=[a] are a, a^3,a^5, a^7, and the generators of G'= are b, b^3, b^5,b^7, then to me it looks more logical to have these isomorphisms:
first [tex]\theta(a)=b^i; \theta(a^3)=b^i, \theta(a^5)=b^i;\theta(a^7)=b^i,i=1,3,5,7[/tex] , which means not 4, but 4! isomorphisms.

So, what am i missing here?
 
  • #4
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Once you map a generator somewhere you fix where every other element in the group goes. You can't map say a to b and a^3 to b^5 with one homomorphism because the first implies that a^3 must map to b^3.
 
  • #5
1,631
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well, yeah, i think i get it.

thnx
 

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