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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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    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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Hint: Considering $\Bbb Z_4=\{0,1,2,3\}$ with $+$, we'll have that $\phi(1)$, the image of $1$, totally describes all the homomorphism $\phi$. What can $\phi(1)$ be?

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    The possible solutions for $\phi(1)$ are $0,2,4,6$, these are the ones that satisfy $4x=0$ in $\Bbb Z_8$.2012-11-28