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I am wondering if anyone could help me with a question I have. The question states: "Describe all homomorphisms from the group $\Bbb Z_4$ to the group $\Bbb Z_8$. "

I'm not sure where to start.

Thanks!

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    Start with the definition of homomorphism. What does this definition allow you to say about what the homomorphisms could and could not be?2012-11-27
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    A homomorphism is a function that transforms one group to another group. So in this case, I start with Z_4 and go to Z_8. I believe that means that there exists a function that maps Z_4 to Z_8. I also know that if such a function exists then it means that f(ab) = f(a)f(b). What elements in Z_4 are mapped to Z_8 though? This is where I am getting stuck.2012-11-28
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    Hey, the *group operation* of $\Bbb Z_4$ and $\Bbb Z_8$ is the **addition**, and not the multiplication!! So, you rather look for $f$ which satisfies $f(a+b)=f(a)+f(b)$.2012-11-28

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