How can the stabilizer of a group element have more than one element itself? I suspect I've misunderstood the concept of a stabilizer.
Take a group $G$ and some $g \in G$. Now consider $s_1, s_2 \in Stab(g)$. The stabilizer is defined such that $s_1 g = s_2 g = g$, but if we operate on the right by $g^{-1}$, we obtain $s_1 = s_2 = e$ where $e$ is the identity. Thus $Stab(g) = \{e\}$.
This is clearly not correct, so I suspect I've failed to fully understand the concept. Where did I go wrong?

The problem is that $g^{-1}g$ is not necessarily $e$! The action of $G$ on itself may be very different from the group law (think about conjugation, for example : in this case, $g^{-1}.g=g^{-1}gg=g\neq e$ in general). 
