Explore BrainMass
Share

Explore BrainMass

    Homomorphisms and Mapping Prime Elements

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

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

    © BrainMass Inc. brainmass.com April 3, 2020, 4:02 pm ad1c9bdddf
    https://brainmass.com/math/linear-transformation/homomorphisms-mapping-prime-elements-97136

    Solution Preview

    The number of such homomorphisms is zero (0).

    ----------

    Why?

    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.

    $2.19

    ADVERTISEMENT