Stabilizers of permutations The problem is I need to list all elements in the stabilizer of the permutation $\sigma=(123)$ in $S_5$.

I know that the stabilizer of a permutation $\sigma$ is the subgroup $$Stab_{{S_n}}(\sigma) = \left\{ x\in S_n : x\sigma x^{-1}=\sigma \right\}$$

So that I need to find a permutation $x$ such that $$x\sigma x^{-1} = x(123) x^{-1} =(123)$$

Then I used the fact that $$x(123) x^{-1} =(x(1)x(2)x(3)) = (123)$$
Then I realize I'm just getting identities.. am I on the right track? Is it possible to have more than one value of x? Why is it so?
There are definitely more than one element stabilizing $(123)$.  Note that you can rewrite $(123) = (312) = (231)$, which should indicate more possible stabilizers using your method.  

Furthermore, you should check that if $x$ fixes $(123)$ then $x \circ (45)$ also fixes $(123)$.
Hmm. $\text{Stab}(x) = \{g \in G: g \cdot x = x\}$. So we are letting elements in $S_5$ act on $\sigma=(123)$ by regular function composition. Let $\tau \in S_5$ then $\tau \cdot \sigma = \sigma \iff \tau(1)=1,\tau(2)=2$ and $\tau(3)=3$. It would seem as though we could then take any permutation of the form $\tau=(xy)$ where $x,y = 4$ or $5$. This is how you get $(45)$. Now do this for $(312)$ and $(231)$.