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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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    [This](https://math.stackexchange.com/questions/1807358/number-of-homomorphisms-between-two-cyclic-groups) is a related post.2017-07-10

1 Answers 1

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Hint:

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


Edit:

Your thoughts on $(a)$ are indeed correct.

Use similar reasoning, along with the given hint to arrive at answers for $(b)$ and $(c)$. See what you can do, and I'll be happy to follow up in comments.


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    @Kns, there are only 2 for (c), i.e. $f(1)=[5]$ and $f(1)=[0]$. The only possibilities to define $f:\mathbb{Z}_n \to \mathbb{Z}_m$ on a generator is to map $1$ to a multiple of $\frac{m}{gcd(m,n)}$. (Try to figure out why). There are thus $gcd(m,n)$ possibilities.2016-06-08