Originally Posted by

**Jhevon** There are only 2.

define $\displaystyle \theta_1(x) = x$ for $\displaystyle x \in \mathbb{Z}

$

and $\displaystyle \theta_2 (x) = -x$ for $\displaystyle x \in \mathbb{Z}$

check that these are homomorphisms

also, they will be the only ones. since any homomorphism has to be of the form $\displaystyle \theta (x) = nx$ for some integer n. this only works with n = +/- 1, otherwise the map would not be onto.

from directly above, we can choose n to be any integer here, so there are an infinite amount.

there are 2.