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Homework Help: Question about groups ?

  1. Jul 28, 2011 #1
    1. The problem statement, all variables and given/known data
    Just want to make something clear. Are all cyclic groups that have the same number of elements isomorphic to each other.

    3. The attempt at a solution
    I think yes because theirs is a one-to-one correspondence and the groups are cyclic which means they have generators.
     
  2. jcsd
  3. Jul 28, 2011 #2

    HallsofIvy

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    Yes, all cyclic groups of order n are isomorphic. You can define the isomorphism itself by mapping the generator of one group to the generator of the other group.
     
  4. Jul 28, 2011 #3
    ok thanks for your answer, can we have non-cyclic groups be isomorphic to each other.
     
  5. Jul 28, 2011 #4
    Does an isomorphism between groups require that the groups be cyclic? No. Take the additive group of real numbers and the multiplicative group of positive reals. They are isomorphic via [itex] x \mapsto e^x[/itex].

    However, remember that if G and G' are isomorphic groups, then G is cyclic if and only if G' is.
     
  6. Jul 28, 2011 #5
    ok thanks for your answer
     
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