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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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Solution Preview

The number of such homomorphisms is zero (0).

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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.

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