# Locally elusive classical groups

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## Abstract

Let *G* be a transitive permutation group of degree n with point stabiliser *H* and let *r* be a prime divisor of *n*. We say that *G* is *r*-elusive if it does not contain a derangement of order *r*. The problem of determining the *r*-elusive primitive groups can be reduced to the almost simple case, and the purpose of this paper is to complete the study of *r*-elusivity for almost simple classical groups. Building on our earlier work for geometric actions of classical groups, in this paper we handle the remaining nongeometric actions where *H* is almost simple and irreducible. This requires a completely different approach, using tools from the representation theory of quasisimple groups.

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## References

- [1]M. Aschbacher, On the maximal subgroups of the finite classical groups, Inventiones Mathematicae
**76**(1984), 469–514.Google Scholar - [2]M. Aschbacher and G. M. Seitz, Involutions in Chevalley groups over fields of even order, Nagoya Mathematical Journal
**63**(1976), 1–91.Google Scholar - [3]N. Bourbaki, Lie Groups and Lie Algebras (Chapters 4–6), Elements of Mathematics, Springer, Berlin, 2002.Google Scholar
- [4]J. N. Bray, D. F. Holt and C. M. Roney-Dougal, The Maximal Subgroups of the Lowdimensional Finite Classical Groups, London Mathematical Society Lecture Note Series, Vol. 407, Cambridge University Press, Cambridge, 2013.Google Scholar
- [5]T. C. Burness, Fixed point spaces in primitive actions of simple algebraic groups, Journal of Algebra
**265**(2003), 744–771.Google Scholar - [6]T. C. Burness, Fixed point spaces in actions of classical algebraic groups, Journal of Group Theory
**7**(2004), 311–346.Google Scholar - [7]T. C. Burness, 3, Journal of Algebra
**309**(2007), 80–138.Google Scholar - [8]T. C. Burness, Fixed point ratios in actions of finite classical groups, IV, Journal of Algebra
**314**(2007), 749–788.Google Scholar - [9]T. C. Burness and M. Giudici, Classical Groups, Derangements and Primes, Australian Mathematical Society Lecture Series, Vol. 25, Cambridge University Press, Cambridge, 2016.Google Scholar
- [10]T. C. Burness, M. Giudici and R. A. Wilson, Prime order derangements in primitive permutation groups, Journal of Algebra
**341**(2011), 158–178.Google Scholar - [11]P. J. Cameron, M. Giudici, G. A Jones, W. M. Kantor, M. H. Klin, D. Marušič and L. A. Nowitz, Transitive permutation groups without semiregular subgroups, Journal of the London Mathematical Society
**66**(2002), 325–333.Google Scholar - [12]B. Fein, W. M. Kantor and M. Schacher, Relative Brauer groups II, Journal für die Reine und Angewandte Mathematik
**328**(1981), 39–57.Google Scholar - [13]The GAP Group, GAP—Groups, Algorithms and Programming, Version 4.4, 2004.Google Scholar
- [14]M. Giudici,
*Quasiprimitive groups with no fixed point free elements of prime order*, Journal of the London Mathematical Society**67**(2003), 73–84.Google Scholar - [15]M. Giudici and S. Kelly, Characterizing a family of elusive permutation groups, Journal of Group Theory
**12**(2009), 95–105.Google Scholar - [16]M. Giudici, L. Morgan, P. Potočnik and G. Verret, Elusive groups of automorphisms of digraphs of small valency, European Journal of Combinatorics
**46**(2015), 1–9.Google Scholar - [17]D. Gorenstein, R. Lyons and R. Solomon, The Classification of the Finite Simple Groups, Number 3, Mathematical Surveys and Monographs, Vol. 40, American Mathematical Society, Providence, RI, 1998.Google Scholar
- [18]R. M. Guralnick and J. Saxl, Generation of finite almost simple groups by conjugates, Journal of Algebra
**268**(2003), 519–571.Google Scholar - [19]G. Hiss and G. Malle, Low dimensional representations of quasi-simple groups, LMS Journal of Computation and Mathematics
**4**(2001), 22–63.Google Scholar - [20]G. Hiss and G. Malle, Corrigenda: Low dimensional representations of quasi-simple groups, LMS Journal of Computation and Mathematics
**5**(2002), 95–126.Google Scholar - [21]J. E. Humphreys, Conjugacy Classes in Semisimple Algebraic Groups, Mathematical Surveys and Monographs, Vol. 43, American Mathematical Society, Providence, RI, 1995.Google Scholar
- [22]G. D. James, On the minimal dimensions of irreducible representations of symmetric groups, Mathematical Proceedings of the Cambridge Philosophical Society
**94**(1983), 417–424.Google Scholar - [23]C. Jansen,
*The minimal degrees of faithful representations of the sporadic simple groups and their covering groups*, LMS Journal of Computation and Mathematics**8**(2005), 122–144.Google Scholar - [24]W. M. Kantor and Á. Seress, Prime power graphs for groups of Lie type, Journal of Algebra
**247**(2002), 370–434.Google Scholar - [25]P. B. Kleidman and M. W. Liebeck, The Subgroup Structure of the Finite Classical Groups, London Mathematical Society Lecture Note Series, Vol. 129, Cambridge University Press, Cambridge, 1990.Google Scholar
- [26]A. S. Kleshchev and P. H. Tiep, Small-dimensional projective representations of symmetric and alternating groups, Algebra & Number Theory
**6**(2012), 1773–1816.Google Scholar - [27]V. Landazuri and G. M. Seitz, On the minimal degrees of projective representations of the finite Chevalley groups, Journal of Algebra
**32**(1974), 418–443.Google Scholar - [28]M. W. Liebeck and G. M. Seitz, Unipotent and nilpotent classes in simple algebraic groups and Lie algebras, Mathematical Surveys and Monographs, Vol. 180, American Mathematical Society, Providence, RI, 2012.Google Scholar
- [29]M. W. Liebeck and G. M. Seitz,
*Reductive subgroups of exceptional algebraic groups*, Memoirs of the American Mathematical Society 121 (1996).Google Scholar - [30]M. W. Liebeck and A. Shalev,
*Simple groups, permutation groups and probability*, Journal of the American Mathematical Society**12**(1999), 497–520.Google Scholar - [31]F. Lübeck,
*Small degree representations of finite Chevalley groups in defining characteristic*, LMS Journal of Computation and Mathematics**4**(2001), 135–169.Google Scholar - [32]R. P. Martineau,
*On 2-modular representations of the Suzuki groups*, American Journal of Mathematics**94**(1972), 55–72.Google Scholar - [33]C. E. Praeger, Á, Seress and Ş. Yalçinkaya,
*Generation of finite classical groups by pairs of elements with large fixed point spaces*, Journal of Algebra**421**(2015), 56–101.Google Scholar - [34]J.-P. Serre, On a theorem of Jordan, Bulletin of the American Mathematical Society
**40**(2003), 429–440.Google Scholar - [35]R. Steinberg,
*Endomorphisms of linear algebraic groups*, Memoirs of the American Mathematical Society 80 (1968).Google Scholar - [36]G. E. Wall,
*On the conjugacy classes in the unitary, symplectic and orthogonal groups*, Journal of the Australian Mathematical Society**3**(1963), 1–62.Google Scholar - [37]J. Xu,
*On elusive permutation groups of square-free degree*, Communications in Algebra**37**(2009), 3200–3206.Google Scholar

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