How can we define that function?

Question

I want to show that $M=\{\tau \in S_4\mid \tau (4)=4\}$ is isomorphic to $S_3$.

To do that we have to consider a function $f(x)$ that gives the isomorphism of $M$ with $S_3$, i.e., we have to describe what $f(x)$ is in $S_3$ for each $x\in M$, right?

We have that $f(\tau)\in \{1,2,3\}$ for $\tau \in S_3$ and $f(\tau (4))=4$, right?

How can we define that function?

Answer 1

An element of $S_3$ is a map from $\{1,2,3\}$ to itself. If we are given $\tau\in M$ then a priori this is a map from $\{1,2,3,4\}$ to itself. Of course we can always restrict such a map to a subset, thus obtaining a mpa $\{1,2,3\}\to\{1,2,3,4\}$. But by the given condition that $\tau(4)=4$ (and hence $\tau(x)\ne 4$ for $x\ne 4$) we see that we can consider this restriction as a map $\{1,2,3\}\to\{1,2,3\}$ (and a bijective map of course) as desired.

Answer 2

Given $\sigma\in\ S_3$ define $f(\sigma)$ to be the permutation in $S_4$ fixing $4$ and agreeing with $\sigma$ on $1$, $2$ and $3$.