Let $S_\omega$ denote the collection of bijections $f:\omega\to\omega$. We say that $f \in S_\omega$ has a fixed point if there is $x\in \omega$ with $f(x) = x$.

It is a short exercise to show that if $f,g\in S_\omega$ then $g\circ f$ has a fixed point if and only if $f\circ g$ has a fixed point.

Let $$E = \big\{\{f,g\}: f\neq g \in S_\omega \text{ and } f\circ g \text{ has a fixed point}\big\}.$$

Question. If $G$ is a finite graph, is $G$ isomorphic to an induced subgraph of $(S_\omega, E)$?