Question

How many surjective homomorphisms are there from `ZZ` onto `ZZ_3` ?

Answer 1

There are two.

Explanation:

Assuming we are talking about `ZZ` and `ZZ_3` as additive groups, there are exactly two surjective homomorphisms from `ZZ` onto `ZZ_3`.

Suppose `f:ZZ -> ZZ_3` is a surjective homomorphism.

Then the identity must map to the identity, i.e. `f(0) = hat(0)`

The value of `f(n)` for any other `n` is determined by the value of `f(1)`. The mapping `f(1) = hat(0)` yields a homomorphism, but it is not surjective: every integer is mapped to `hat(0)`.

That leaves two possibilities, namely `f(1) = hat(1)` and `f(1) = hat(2)`. Both of these result in surjective homomorphisms.