Ways $S_3$ can act on a set of 4 elements. Describe all ways in which $S_3$ can operate on a set of four elements.
My approach: This question can be broken down into: How many homomorphisms exist from $S_3$ to $S_4$. Say $\varphi : S_3 \to S_4$ is a homomorphism. Then we have three possibilities for $\text{ker }\varphi$: $\{1\}, \{1, (1\ 2\ 3), (1\ 3\ 2)\}$, and $S_3$.
The case in which $\text{ker }\varphi = S_3$ is the trivial homomorphism that maps everything to the identity.
Now, the case in which $\text{ker }\varphi = \{1\}$ is the same as saying that the mappings are injective. This comes down to picking three of the four elements and permutating them and leaving the fourth one fixed. There are $\binom43 = 4$ ways of doing this.
Say $\text{ker }\varphi = \{1, (1\ 2\ 3), (1\ 3\ 2)\}$. This means that $\varphi((1)) = \varphi((1\ 2\ 3)) = \varphi((1\ 3\ 2)) = (1)$. Morover we can observe the following two properties immediately:


*

*$\varphi((1\ 2\ 3)) = \varphi((1\ 3))\varphi((1\ 2)) = (1)$. Equivalently $\varphi((1\ 2)) = \varphi((1\ 3))$.

*$\varphi((1\ 3\ 2)) = \varphi((1\ 3))\varphi((2\ 3)) = (1)$. Equivalently $\varphi((1\ 3)) = \varphi((2\ 3))$.


and thus $\varphi((1\ 2)) = \varphi((1\ 3)) = \varphi((2\ 3))$. But we know by properties of homomorphisms that $\vert \varphi((1\ 3)) \vert \mid \vert (1\ 3) \vert = 2$. So $\vert \varphi((1\ 3)) \vert$ is 1 or 2. But if the order of $\varphi((1\ 3))$ were 1 it would be in the kernel, which would be a contradiction to the kernel we chose, so it must be 2. We can map $(1\ 3)$ to any 2-cycle in $S_4$, of which there are 6, as well as any product of disjoint 2-cycles, of which there are 3. Hence we have 9 possible homomorphic mappings given this kernel.
Adding up all the possible homomorphisms from $S_3$ to $S_4$ that we counted, we get 14 different ways in which $S_3$ can act on four elements, as described above.
Is this correct? Are there 14 homomorphisms from $S_3$ to $S_4$? Is my reasoning correct? Or are there any hidden assumptions I made that I shouldn't have made?
 A: For the "medium-sized" kernel $\{1,(1\,2\,3),(1\,3\,2)\}$, you might simply observer that such an operation is indeed equivalent to a faithful operation $S_3/\ker \phi\cong S_2$. And yes, such an operation is given exectly by choosing an element of order two in $S_4$.
However, I am not satisfied with your argument for trivial kernel, i.e. that an injection $S_3\to S_4$ automatically means that one point is fixed. For example, with six instead of four elements this would obviously not be true: One way to operate would be to perform the same permutations on the first and on thel last three elements.
With four elements as actually given here, you are right, but it has to be shown! (Hint: Consider the operation of a 3-cycle.)
Moreover, merely selecting three out four elements does not uniquely determine the operation. Ultimately, you should find 24 instead of 4 faithful operations.

After so many comments let's start afresh counting all homomorphisms $f\colon S_3\to S_4$.
Note that $S_3$ is generated by $\sigma =(1\,2)$ and $\tau=(1\,2\,3)$, hence $f$ is already determined if we know $f(\sigma)$ and $f(\tau)$.
The order of $f(\tau)$ must be a divisor of the order of $\tau$, i.e. either $f(\tau)=1$ or $f(\tau)$ is one of the 8 elements of order 3 in $S_4$. Similarly, $f(\sigma)=1$ or it is one of the 9 elements of order 2.
(i) If $f(\tau)=1$ nothing restricts our choice for $f(\sigma)$, so we find $10$ homomorphisms of this kind.
(ii) If $f(\tau)$ is of order three, say $f(\tau)=(a\,b\,c)$, then our choice for $f(\sigma)$ is somewhat restricted because $\sigma\tau=(2\,3)$ is of order 2. This directly forbids $f(\sigma)=1$. If $f(\sigma)$ affects the fourth element $d$, then wlog. $f(\sigma)(d)=a$ and hence $f(\sigma\tau)(d)=b$. Therefore we must have $f(\sigma\tau)(b)=d$, i.e. $f(\sigma)(c)=d$. But if $f(\sigma)$ permutes $c\mapsto d\mapsto a$, it cannot be of order 2. Hence $f(\sigma)$ merely is a permutation of order two of the set $\{a,b,c\}$. Wlog. $f(\sigma)=(a\,b)$. We have 4 choices for $d$ (the fixpoint) and then 3 choices for $c$ (the other fixpoint of $f(\sigma)$), two choices for $a$ (the image of $c$ under $f(\tau)$). (And in fact any such choice is valid: We simply replace the elements $1, 2, 3$ with $a, b, c$ in that order and leave $d$ fixed, thus this is essentially the canonical operation of $S_3$ on $\{1,2,3\}$). Thus there are exactly $24$ homomorphisms of this kind.
A: We work instead in terms of the corresponding permutations representations (homomorphisms) $\phi: S_3\to S_4$. To avoid confusion we work with $S_3(\{1,2,3\})$ and $S_4(\{a,b,c,d\})$. Let $x = (1,2)$ and $y = (1,2,3)$, so that $$x^2=y^3=1,\text{ }y^2 = (1,3,2),\text{ }xy = (2,3) = yx^2,\text{ }xy^2 = (1,3) = yx.$$
By the First Isomorphism Theorem, $\text{Ker}\,\phi$ is normal in $S_3$ and $\text{Im}\,\phi \cong S_3/\text{Ker}\,\phi$, with isomorphism $\phi(g\text{Ker}\,\phi) = \phi(g)$. But$$yxy^{-1} = xy^2y^{-1} = xy\ne 1,\,x,$$so $S_3$ has no normal subgroups of order $2$, and $\text{Ker}\,\phi$ must be one of $\{1\}$, $A_3$, $S_3$.
If $\text{Ker}\,\phi = S_3$, then$$\text{Im}\,\phi\cong S_3/S_3\cong S_1,$$so $\phi$ is the trivial homomorphism, i.e. it always maps to $(a)(b)(c)(d)$.
If $\text{Ker}\,\phi = A_3$, then$$\text{Im}\,\phi\cong S_3/A_3\cong S_2,$$ $$\phi(1)=\phi(y)=\phi(y^2) = (a)(b)(c)(d),$$while$$\phi(x)=\phi(xy)=\phi(xy^2)$$ has order $2$ and takes the form $(p,q)$ or $(p,q)(r,s)$, for distinct $p,q,r,s\in\{a,b,c,d\}$.
If $\text{Ker}\,\phi = \{1\}$, then$$\text{Im}\,\phi\cong S_3/\{1\} \cong S_3.$$So $\phi(x)$ takes the form $(p,q)$ or $(p,q)(r,s)$, and $\phi(xy)$ takes the form $(p',q')$ or $(p',q')(r',s')$. Without loss of generality, $p' = p$.


*

*If $\phi(x) = (p,q)$ and $\phi(xy) = (p,q')$, then $q\ne q'$, or else$$\phi(y) = \phi(x)\phi(xy) = (p,q)(p,q')$$becomes the identity, so $\phi(y) = (q,p,q')$. Because $\phi(x) = (q,p)$, $\phi$ simply maps $\pi\in S_3(\{1,2,3\})$ to the corresponding permutation with $1$, $2$, $3$ replaced by $q$, $p$, $q'$, respectively.

*If $\phi(x) = (p,q)$ and $\phi(xy) = (p,q')(r',s')$ (or the other way around; these two cases are interchangeable by "switching" $1$ and $3$, since $x$, $xy$ are both transpositions), then $q\ne q'$, or else$$\phi(y) = \phi(x)\phi(xy) = (p,q)(p,q')(r',s')$$becomes a transposition and has order $2$ instead of $3$, so without loss of generality, assume $r' = q$. But then$$\phi(y) = (q,p,q')(q,s') = (q,s',p,q')$$has order $4$ instead of $3$, so this case is impossible.

*If $\phi(x) = (p,q)(r,s)$ and $\phi(xy) = (p,q')(r',s')$, then $q\ne q'$, or else $\{r,s\} = \{r',s'\}$ and $\phi(x)=\phi(xy)$ forces $\phi(y)$ to be the identity. Without loss of generality, assume $q' = r$, so $\{r',s'\} = \{q,s\}$ means $\phi(xy) = (p,r)(q,s)$. Thus$$\phi(y) = (p,q)(r,s)(p,r)(q,s) = (p,s)(q,r)$$ has order $2$ instead of $3$, so this case is also impossible.


Remark. The only "unexpected" permutation here is $\phi$ taking transpositions to $(a, b)(c, d)$ (or $(a, c)(b, d)$ or $(a, d)(b, c)$) and other elements to the identity $(a)(b)(c)(d)$.
