How many homomorphisms are there of $\mathbb{Z}/{3\mathbb{Z}} $ into $S_3$?

Studying groups of order $12$, specifically, the semi-direct products that appear in these groups, I come across something that I do not fully understand.

In Dummit (pag. 183 groups order 12) you need to see the amount of homomorphisms that there between $T\simeq \mathbb{Z}/{3\mathbb{Z}}$ and ${\rm Aut}(V)\simeq S_3$. The argument they give is the following:

$\mathbb{Z}/{3\mathbb{Z}}$ is cyclic. Then let $T=\langle y\rangle $and there is a unique subgroup of ${\rm Aut}(V)$ of order $3$, say $\langle \gamma\rangle$ . This, there are three possible homomorphisms from $T$ into ${\rm Aut}(V)$:

$\varphi_i:T\to{\rm Aut}(V)$ defined by $\varphi_i(y)=\gamma^i$, $i=0,1,2.$

Question: Why are there only three possible homomorphisms from $T$ into ${\rm Aut}(V)$?

I know it has to do with the order of the domain of the homomorphism but I don't see why the possible homomorphisms need to reach a subgroup with the same order of their domain (for the homomorphism to be well defined)

Solution 1:

Say $G$ is cyclic of order $n$ with generator $g$ and $T \colon G \to K$ a group morphism. Then $T(g)^n = T(g^n) = T(1) = 1$, so the order of $T(g)$ divides $n$. Likewise, if $x \in K$ has order divisible by $n$, you can check that $T(g^k) := x^k$ yields a morphism.

Hence $\hom_{\mathsf{Grp}}(\mathbb{Z}/n\mathbb{Z}, K)$ is in correspondence with the elements of $K$ whose order is divisible by $n$. When $n=3$ there are not many options: these are precisely the elements of order $1$ or $3$.

Now, an element $x$ of order $3$ generates a subgroup $\langle x\rangle = \{1,x,x^2\}$ of order three. If there is only one such group in $K$, then all elements of order three are just $x$ and $x^2$. Together with $1$, they determine all maps $\mathbb{Z}/3\mathbb{Z} \to K$.

Solution 2:

Note that maps $\mathbb{Z} \to G$ correspond exactly to elements of $G$ - $f$ corresponds to $f(1)$. Note that maps $\mathbb{Z} / 3\mathbb{Z} \to G$ correspond exactly to maps $f : \mathbb{Z} \to G$ such that $3 \mathbb{Z} \subseteq \ker(f)$. And note that $3 \mathbb{Z} \subseteq \ker(f)$ iff $f(3) = e$. So maps $\mathbb{Z} / 3 \mathbb{Z} \to G$ correspond to elements $x \in G$ such that $x^3 = e$.

Thus, the real question is: what automorphisms $f \in {\rm Aut}(V)$ are there such that $f^3 = e$? Or equivalently, we must find $f$ such that the order of $f$ is a factor of 3 - that is, such that the order is either 1 or 3. There are exactly 3 such maps.