- #1

ArcanaNoir

- 779

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## Homework Statement

Describe all group homomorphisms from [itex] \mathbb{Z}_n [/itex] to [itex] \mathbb{Z}_m [/itex].

## Homework Equations

[itex] \mathbb{Z}_n = {[0],[1],\dots ,[n-1]} [/itex] with addition.

A homomorphism is an operation preserving map, ie [itex] \phi (a\ast b)=\phi (a) \# \phi (b) [/itex].

One especially important homomorphism property is that [itex] \phi (a^k) = \phi (a)^k [/itex].

We can describe each homomorphism entirely by its action on any element that generate the group.

## The Attempt at a Solution

I am pretty sure there are [itex] \text{gcd}(n,m) [/itex] homomorphisms from [itex] \mathbb{Z}_n [/itex] to [itex] \mathbb{Z}_m [/itex].

Based on some examples I worked out, I believe the solution is:

let [itex] [a] [/itex] be any element which generates [itex] \mathbb{Z}_n [/itex]

[tex] \phi ([a]) = \frac{n}{\text{gcd}(n,m)}\cdot k [a] [/tex] where [itex] 0\le k < \text{gcd}(m,n) [/itex]