# Z-Modules and Isomorphisms

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(8) Prove that Hom_Z(Z/nZ,Z/mZ) is isomorphic to Z/(n,m)Z

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#### Solution Preview

Proof:

We know, Z/nZ={0,1,...,n-1}, Z/mZ={0,1,...,m-1}. Let d=(n,m), then

Z/dZ={0,1,...,d-1}. As a group, Z/nZ,Z/mZ,Z/dZ are all cyclic groups.

So each homomorphism f from Z/nZ to Z/mZ is uniquely determined by f(1).

Let f_k in ...

#### Solution Summary

Z-Modules and Isomorphisms are investigated.

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