[Math] How to find the group homomorphisms from $\Bbb Z$ to $D_4$

Question

I am trying to find the homomorphisms from $\Bbb Z$ to $D_4$. I know that I need to start by finding the normal subgroups of $\Bbb Z$ that could be the kernel of the homomorphism, but am not sure how to proceed from there.

Answer 1

Since the image of $\Bbb Z$ is a cyclic group, you need only consider the cyclic subgroups of $D_4$--where you map $1$ to completely determines the image. Now here things are a bit tricky since some people define $D_n$ to be the dihedral group of order $2n$ and others the one of order $2n$.

In the former case, you have $2$ homomorphisms onto the cyclic subgroup of order $4$ (since there are two generators) and $5$ more into cyclic subgroups of order $2$ (since all the other elements have order $2$ because the cyclic subgroup of $D_n$ is characteristic). Add in the trivial homomorphism for a total of 8.

In the latter case your group has order $4$ and is isomorphic to the Klein-$4$ group, so you have three homomorphisms, one onto each of the subgroups of order $2$, plus the trivial homomorphism for a total of four.

From working out both of these we see a pattern emerge: send $1\in\Bbb Z$ to anything and it automatically defines a homomorphism, so in any group there are exactly $|G|$ homomorphisms from $\Bbb Z$ to $G$: one for each element of $G$ defined by $\phi_g: 1\mapsto g$.