# Find the number of homomorphisms

In each of the following examples determine the number of homomorphisms between the given groups:

$(a)$ from $\mathbb{Z}$ to $\mathbb{Z}_{10}$;

$(b)$ from $\mathbb{Z}_{10}$ to $\mathbb{Z}_{10}$;

$(c)$ from $\mathbb{Z}_{8}$ to $\mathbb{Z}_{10}$.

Could anyone just give me hints for the problem? Well, let $f:\mathbb{Z}\rightarrow \mathbb{Z}_{10}$ be homo, then $f(1)=[n]$ for any $[n]\in \mathbb{Z}_{10}$ will give a homomorphism hence there are $10$ for (a)?

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1
Yes, that's exactly right. f(1) is all that matters, because 1 generates the whole group. – Billy Dec 17 '12 at 20:49

## 1 Answer

Hint:

A homomorphism on a cyclic group is completely determined by its value on a generator of the group.

Edit:

You're thoughts on $(a)$ are indeed correct.

Can you apply similar reasoning to arrive at answers for $(b)$ and $(c)$?

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Is it 10 for (b) and (c) as well? – Vivek Dec 18 '12 at 13:40
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