Distinguishing simple groups

The distinguishing number $D(\mathrm{\Gamma })$ of a graph $\mathrm{\Gamma }$ is the least size of a partition of the vertices of $\mathrm{\Gamma }$ such that no non-trivial automorphism of $\mathrm{\Gamma }$ preserves this partition. We show that if the automorphism group of a graph $\mathrm{\Gamma }$ is simple, than $D(\mathrm{\Gamma })=2$. This is obtained by establishing the distinguishing number for all possible actions of simple groups.