The distinguishing number of quasiprimitive and semiprimitive groups

Abstract

The distinguishing number of $$G \leqslant \mathrm {Sym}(\Omega )$$ is the smallest size of a partition of $$\Omega $$ such that only the identity of G fixes all the parts of the partition. Extending earlier results of Cameron, Neumann, Saxl, and Seress on the distinguishing number of finite primitive groups, we show that all imprimitive quasiprimitive groups have distinguishing number two, and all non-quasiprimitive semiprimitive groups have distinguishing number two, except for $$\mathrm {GL}(2,3)$$ acting on the eight non-zero vectors of $$\mathbb {F}_3^2$$ , which has distinguishing number three.