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 March 4, 2021, 7:22 pm ad1c9bdddf
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 ...
Homomorphisms and Mapping Prime Elements are investigated.