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I have a question, which may be trivial, but I really don't understand how I should solve the next exercise:

List all group endomorphisms of $\mathbb{Z}_6$.

I consider that I must write down all the functions $f\colon \mathbb{Z}_6\to\mathbb{Z}_6$. But which is the general form of the function?

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    I know what an endomorphism of a group G is: the function f:(G,+)→(G,+), with f(a+b)=f(a)+f(b), a,b from G; and f(0)=0. But in my case, I still don't know which is the general form of the function.2012-02-03

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Not all functions, but only those that are additive homomorphisms. That is, only functions $f\colon\mathbb{Z}_6\to\mathbb{Z}_6$ such that $f(a+b) = f(a)+f(b)$ for all $a,b\in\mathbb{Z}_6$.

  1. Show that if $f\colon\mathbb{Z}_6\to\mathbb{Z}_6$ is an additive homomorphism, and you know what $f(1)$ is, then you know what $f(a)$ is for all $a\in\mathbb{Z}$.

  2. Determine what values $f(1)$ can be to give you an additive homomorphism.