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The distinguishing number of quasiprimitive and semiprimitive groups

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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.

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Acknowledgements

The authors thank Gabriel Verret for highlighting this problem and the Centre for the Mathematics of Symmetry and Computation, where this work began at the 2018 Research Retreat. The second author is grateful for the Cecil King Travel Scholarship from the LMS and the hospitality of the University of Western Australia; he also thanks EPSRC and the Heilbronn Institute for Mathematical Research for their financial support. The third author gratefully acknowledges the support of the ARC grant DE160100081.

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  1. Luke Morgan

    Present address: University of Primorska, UP FAMNIT, Glagolja ska 8, 6000, Koper, Slovenia

Authors and Affiliations

  1. Department of Mathematics and Statistics (M019) Centre for the Mathematics of Symmetry and Computation, The University of Western Australia, 35 Stirling Highway, Crawley, WA, 6009, Australia

    Alice Devillers

  2. University of Primorska, UP IAM, Muzejski trg 2, 6000, Koper, Slovenia

    Luke Morgan

  3. School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK

    Scott Harper

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  1. Alice Devillers
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  2. Luke Morgan
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Correspondence to Luke Morgan.

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Devillers, A., Morgan, L. & Harper, S. The distinguishing number of quasiprimitive and semiprimitive groups. Arch. Math. 113, 127–139 (2019). https://doi.org/10.1007/s00013-019-01324-7

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