Is there a nontrivial action of $\mathbb{Z}/3$ on $\mathbb{Z}$ as automorphisms?

Question

So I've started to learn some algebra and group actions. There was an example of how $\mathbb{Z}/2$ can act on $\mathbb{Z}$ as $n\mapsto n$ or $n\mapsto -n$. Are there any other examples of nontrivial action of $\mathbb{Z}/3$ (or more generally $\mathbb{Z}/p$) on $\mathbb{Z}$?

Answer 1

To begin with, let's think about the apparently-harder problem of classifying all automorphisms of $\mathbb{Z}$. In this case actually it's quite easy - there are exactly two, the trivial one and $n\mapsto -n$ (why? think about the possible targets for $1$ under an arbitrary automorphism ...). So $Aut(\mathbb{Z})$ is isomorphic to $\mathbb{Z}/2$.

Now, this isn't exactly what you asked, but there is an important connection: an action of $G$ on $\mathcal{X}$ by automorphisms is just a homomorphism from $G$ to $Aut(\mathcal{X})$! Just look at what each definition says ... (HINT: It may be easier to first convince yourself that an arbitrary action of a group $G$ on a set $X$ is just a homomorphism from $G$ to the group of all permutations of $X$.)

So to understand the ways $\mathbb{Z}/3$ can act on $\mathbb{Z}$, it's enough to understand the homomorphisms from $\mathbb{Z}/3$ to $Aut(\mathbb{Z})=\mathbb{Z}/2$. This is a much more concrete problem:

• Do you see why there is no nontrivial homomorphism from $\mathbb{Z}/3$ to $\mathbb{Z}/2$? (HINT: similarly to the argument that $Aut(\mathbb{Z})=\mathbb{Z}/2$, think about the behavior of a putative homomorphism on a generator of $\mathbb{Z}/3$. More generally, to determine the set of homomorphisms from $G_1$ to $G_2$ we think about the possible actions of a homomorphism on a generating set for $G_1$.)

• What does this imply about actions of $\mathbb{Z}/3$ on $\mathbb{Z}$ by automorphisms?