# Homomorphisms and Mapping Prime Elements

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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