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    Homomorphisms and Mapping Prime Elements

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    Let G and H be two finite cyclic groups of relatively prime orders. Determine the number of homomorphisms from G to H.

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    The number of such homomorphisms is zero (0).



    Reason by intuition: You should smell something wrong or at least
    your antennas should vibrate whenever there are two cyclic groups
    of unrelated orders. How to confirm your instincts?

    Proof by contradiction:-
    Suppose G has m elements and identity element e.
    Suppose H has n elements and identity element 1.

    Suppose there were such a thing as a homomorphism
    f : G ----> H
    Let g be a generator for G and h be the image of g under
    this purported homomorphism.
    f : e |---> 1
    f : g ...

    Solution Summary

    Homomorphisms and Mapping Prime Elements are investigated.