Finite primitive groups and regular orbits of group elements

Guest, Simon; Spiga, Pablo

doi:10.1090/tran6678

Finite primitive groups and regular orbits of group elements
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by Simon Guest and Pablo Spiga PDF

Trans. Amer. Math. Soc. 369 (2017), 997-1024 Request permission

Abstract:

We prove that if $G$ is a finite primitive permutation group and if $g$ is an element of $G$, then either $g$ has a cycle of length equal to its order, or for some $r$, $m$ and $k$, the group $G \leq \mathrm {Sym}(m) \mathrm {wr}\mathrm {Sym}(r)$ preserves the product structure of $r$ direct copies of the natural action of $\mathrm {Sym}(m)$ on $k$-sets. This gives an answer to a question of Siemons and Zalesski and a solution to a conjecture of Giudici, Praeger and the second author.

References

M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984), no. 3, 469–514. MR 746539, DOI 10.1007/BF01388470
Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235–265. Computational algebra and number theory (London, 1993). MR 1484478, DOI 10.1006/jsco.1996.0125
John N. Bray, Derek F. Holt, and Colva M. Roney-Dougal, The maximal subgroups of the low-dimensional finite classical groups, London Mathematical Society Lecture Note Series, vol. 407, Cambridge University Press, Cambridge, 2013. With a foreword by Martin Liebeck. MR 3098485, DOI 10.1017/CBO9781139192576
Timothy C. Burness, Fixed point ratios in actions of finite classical groups. I, J. Algebra 309 (2007), no. 1, 69–79. MR 2301233, DOI 10.1016/j.jalgebra.2006.05.024
Timothy C. Burness, Fixed point ratios in actions in finite classical groups. II, J. Algebra 309 (2007), no. 1, 80–138. MR 2301234, DOI 10.1016/j.jalgebra.2006.05.025
Timothy C. Burness, Fixed point ratios in actions of finite classical groups. III, J. Algebra 314 (2007), no. 2, 693–748. MR 2344583, DOI 10.1016/j.jalgebra.2007.01.011
Timothy C. Burness, Fixed point ratios in actions of finite classical groups. IV, J. Algebra 314 (2007), no. 2, 749–788. MR 2344584, DOI 10.1016/j.jalgebra.2007.01.012
Timothy C. Burness and Simon Guest, On the uniform spread of almost simple linear groups, Nagoya Math. J. 209 (2013), 35–109. MR 3032138, DOI 10.1017/S0027763000010680
Peter J. Cameron, Finite permutation groups and finite simple groups, Bull. London Math. Soc. 13 (1981), no. 1, 1–22. MR 599634, DOI 10.1112/blms/13.1.1
P. J. Cameron, Projective and polar spaces, Queen Mary and Westfield College Lecture Notes, 2000.
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, $\Bbb {ATLAS}$ of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. MR 827219
L. Emmett and A. E. Zalesski, On regular orbits of elements of classical groups in their permutation representations, Comm. Algebra 39 (2011), no. 9, 3356–3409. MR 2845578, DOI 10.1080/00927872.2010.505936
Daniel Frohardt, Robert Guralnick, and Kay Magaard, Incidence matrices, permutation characters, and the minimal genus of a permutation group, J. Combin. Theory Ser. A 98 (2002), no. 1, 87–105. MR 1897926, DOI 10.1006/jcta.2001.3229
Michael Giudici, Cheryl E. Praeger, and Pablo Spiga, Finite primitive permutation groups and regular cycles of their elements, J. Algebra 421 (2015), 27–55. MR 3272373, DOI 10.1016/j.jalgebra.2014.08.015
D. Gorenstein, R. Lyons, and R. Solomon, The classification of the finite simple groups, Number $3$ Volume 40, 1998.
Simon Guest, Joy Morris, Cheryl E. Praeger, and Pablo Spiga, On the maximum orders of elements of finite almost simple groups and primitive permutation groups, Trans. Amer. Math. Soc. 367 (2015), no. 11, 7665–7694. MR 3391897, DOI 10.1090/S0002-9947-2015-06293-X
Simon Guest, Andrea Previtali, and Pablo Spiga, A remark on the permutation representations afforded by the embeddings of $\textrm {O}_{2m}^\pm (2^f)$ in $\textrm {Sp}_{2m}(2^f)$, Bull. Aust. Math. Soc. 89 (2014), no. 2, 331–336. MR 3182670, DOI 10.1017/S0004972713000828
Robert M. Guralnick and William M. Kantor, Probabilistic generation of finite simple groups, J. Algebra 234 (2000), no. 2, 743–792. Special issue in honor of Helmut Wielandt. MR 1800754, DOI 10.1006/jabr.2000.8357
Peter B. Kleidman, The maximal subgroups of the finite $8$-dimensional orthogonal groups $P\Omega ^+_8(q)$ and of their automorphism groups, J. Algebra 110 (1987), no. 1, 173–242. MR 904187, DOI 10.1016/0021-8693(87)90042-1
Peter Kleidman and Martin Liebeck, The subgroup structure of the finite classical groups, London Mathematical Society Lecture Note Series, vol. 129, Cambridge University Press, Cambridge, 1990. MR 1057341, DOI 10.1017/CBO9780511629235
Maska Law, Alice C. Niemeyer, Cheryl E. Praeger, and Ákos Seress, A reduction algorithm for large-base primitive permutation groups, LMS J. Comput. Math. 9 (2006), 159–173. MR 2221258, DOI 10.1112/S1461157000001236
Martin W. Liebeck and Jan Saxl, Minimal degrees of primitive permutation groups, with an application to monodromy groups of covers of Riemann surfaces, Proc. London Math. Soc. (3) 63 (1991), no. 2, 266–314. MR 1114511, DOI 10.1112/plms/s3-63.2.266
A. V. Vasil′ev and V. D. Mazurov, Minimal permutation representations of finite simple orthogonal groups, Algebra i Logika 33 (1994), no. 6, 603–627, 716 (Russian, with Russian summary); English transl., Algebra and Logic 33 (1994), no. 6, 337–350 (1995). MR 1347262, DOI 10.1007/BF00756348
M. Neunhöffer, Á. Seress: GAP Package “recog” http://www-groups.mcs.st-and.ac.uk/neunhoef.
Guy Robin, Estimation de la fonction de Tchebychef $\theta$ sur le $k$-ième nombre premier et grandes valeurs de la fonction $\omega (n)$ nombre de diviseurs premiers de $n$, Acta Arith. 42 (1983), no. 4, 367–389 (French). MR 736719, DOI 10.4064/aa-42-4-367-389
J. Siemons and A. Zalesskii, Intersections of matrix algebras and permutation representations of $\textrm {PSL}(n,q)$, J. Algebra 226 (2000), no. 1, 451–478. MR 1749899, DOI 10.1006/jabr.1999.8201
Johannes Siemons and Alexandre Zalesskiĭ, Regular orbits of cyclic subgroups in permutation representations of certain simple groups, J. Algebra 256 (2002), no. 2, 611–625. MR 1939125, DOI 10.1016/S0021-8693(02)00107-2