Describe all of the homomorphisms from $\Bbb Z_{20}$ to $\Bbb Z_{40}$.

abstract-algebragroup-homomorphismgroup-theory

I'm starting abstract algebra and I'm having difficulty understanding how to find homomorphism. For instance, I want to find all homomorphisms of $$ \varphi (\Bbb Z_{20}) =\Bbb Z_{40}.$$
I know that the GDC of both groups gives me the number of existing homomorphisms. But I don't know how to proceed and find them all. I know that I'm supposed to take 1 as it is the generator of the group. Could you help me understand how to proceed?

Thank you.

Best Answer

Let the homomorphism as $\varphi$, and $\varphi(1)=x$. Then, $\varphi$ is uniquely determined by $x$.(Why?)

Moreover, $20x=\varphi(20)=\varphi(0)=0$. So, $x$ must be order 20 or its divisor. $x$ can be $0,2,\cdots,38$. So there are 20 homomorphisms(When $x$ is even, then $20x\equiv0\bmod40$).

Then, $\forall n\in\mathbb Z_{20}$, $\varphi(n)=n\varphi(1)=nx.$ (of course, $x=0,2,\cdots,38$.)

Edit

$\varphi$ is additive group homomorphism, so $\varphi(n)=\varphi(1+\cdots+1)=\varphi(1)+\cdots+\varphi(1)=n\varphi(1)$.

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