Probs 4 & 5, Sec 13, in Fraleigh's A FIRST COURSE IN ABSTRACT ALGEBRA, 7th ed: This map of $\mathbb{Z}_m$ to $\mathbb{Z}_n$ is a homomorphism iff ...?

Question

I know that the map $\phi \colon \mathbb{Z}_6 \to \mathbb{Z}_2$, given by $$ \phi(x) = \ \mbox{ the remainder of $x$ when divided by $2$, as in the division algorithm}, $$ is a homomorphism of $\mathbb{Z}_6$ onto $\mathbb{Z}_2$.

However, the map $\phi \colon \mathbb{Z}_9 \to \mathbb{Z}_2$, given by the same formula as above, is not a homomorphism.

Now my question is, given natural numbers $m$ and $n$ such that $m > 1$ and $n > 1$, what is (are) the necessary and sufficient condition(s), or set of conditions, if any, such that the map $\phi \colon \mathbb{Z}_m \to \mathbb{Z}_n$, given by the above formula, is a homomorphism of $\mathbb{Z}_m$ into / onto $\mathbb{Z}_n$?

Answer 1

The map $\mathbb{Z}_9 \to \mathbb{Z}_2$ doesn't even exist! There is a well-defined map $r_n : \mathbb{Z} \to \mathbb{Z}_n$ which sends an integer to its remainder modulo $n$. In fact, $r_n$ is a homomorphism (you should prove this). If you want this to induce a function $\mathbb{Z}_m \to \mathbb{Z}_n$, it better be constant on the equivalence classes $[a] = \{a+km : k\in \mathbb{Z}\}$! So: when is it the case that $r_n(x) = r_n(x+m)$ for all $x \in \mathbb{Z}$?

Answer 2

Hints:

• Regarding an onto homomorphism: note that $\phi:\Bbb Z_m \to \Bbb Z_n$ will be onto if and only if there exists an $x \in \Bbb Z_m$ such that $\phi(x) = 1 \in \Bbb Z_n$. On the other hand, note that for such an $x$ $$ 0 = \phi(0) = \phi(mx) = m\cdot \phi(x). $$

• Regarding an into (one-to-one) homomorphism: note that if $\phi$ is into, then $\phi: \Bbb Z_m \to \Bbb Z_n$ yields an isomorphism and the image of $\phi$, which must be a subgroup of $\Bbb Z_n$.