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Distinguishing simple groups

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