What can you say about groups if every homomorphism is trivial?

Question

Suppose we have two groups $G,H$, and the premise is that every homomorphism from $G$ to $H$ is trivial.

This certainly can happen when either of $G,H$ is trivial, but is it possible that neither of $G,H$ are trivial?

Answer 1

First, remember that there are $(m,n)$ number of homomorphisms between $\mathbb{Z}_n$ and $\mathbb{Z}_m$, see Find the number of homomorphisms between $\mathbb{Z}_m$ and $\mathbb{Z}_n$. Hence if $n$ amd $m$ are coprime, there is only one homomorphism. So it is not necessary for both groups to be trivial.

Answer 2

Yes.

Try $\Bbb Z_2$ and $\Bbb Z_3$.

Answer 3

There are other examples. If $G=\mathbb{Q}$ and $H=\mathbb{Z}$ are the additive group of rational numbers and the additive group of integers respectively, then every homomorphism from $G$ to $H$ is trivial.

Answer 4

By the First homomorphism theorem, this also happens if $G$ is simple and $|H|< |G|$. For example, $A_5\to S_4$.