Math and science::Algebra::Aluffi

Group homomorphisms and order

Let $φ : G \to H$ be a group homomorphism, and let $g \in G$ be an element of finite order. Then $| φ (g) |$ divides $| g |$ .

There are no non-trivial homomorphisms from $Z / n Z$ to $Z$ , as elements of $Z / n Z$ would need to map to elements of finite order in $Z$ , of which there is only the identity, $0$ .
There are no non-trivial homomorphisms from $C_{4}$ to $C_{7}$ , as elements of $C_{7}$ other than the identity all have order 7, which doesn't divide 2 or 4 (the orders of non-identity elements of $C_{4}$ ).
In general, for a non-trivial group morphism $C_{n} \to C_{m}$ to exist, $\gcd (n, m)$ must be greater than 1 as elements of $C_{m}$ must have an order that divides both $| C_{m} |$ and $| C_{n} |$ .

Aluffi, p66