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Math Help - Homomorphism of Groups and their generators

  1. #1
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    Homomorphism of Groups and their generators

    I know that f(g) is a generator of the image of f, say f(G).

    Is it true that given a homomorphism f from a cyclic group G to a goup G' and a generator g of G, f(g) is a generator of G'?

    I think it is positive, but need a concere proof.

    Thank you for considering it.
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  2. #2
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    Quote Originally Posted by smallgun View Post
    Is it true that given a homomorphism f from a cyclic group G to a goup G' and a generator g of G, f(g) is a generator of G'?
    Let \phi : \mathbb{Z} \to \mathbb{Z} be \phi (x) = 0 where \mathbb{Z} are the integers under addition.
    Check that \phi is a homomorphism.
    Not 1 generates \mathbb{Z} but \phi(1) = 0 doth not generate \mathbb{Z}.

    However, if f : G\to G' is an isomorphism then this is true.
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