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  • Cited by 41
Publisher:
Cambridge University Press
Online publication date:
December 2015
Print publication year:
2016
Online ISBN:
9781139059060

Book description

A classical theorem of Jordan states that every finite transitive permutation group contains a derangement. This existence result has interesting and unexpected applications in many areas of mathematics, including graph theory, number theory and topology. Various generalisations have been studied in more recent years, with a particular focus on the existence of derangements with special properties. Written for academic researchers and postgraduate students working in related areas of algebra, this introduction to the finite classical groups features a comprehensive account of the conjugacy and geometry of elements of prime order. The development is tailored towards the study of derangements in finite primitive classical groups; the basic problem is to determine when such a group G contains a derangement of prime order r, for each prime divisor r of the degree of G. This involves a detailed analysis of the conjugacy classes and subgroup structure of the finite classical groups.

Reviews

'This book should be an indispensable reference for anybody doing research, or wanting to do research, in this area. Even for nonspecialists, however, chapters 2 and 3 should be a useful source of information on classical groups.'

Mark Hunacek Source: The Mathematical Gazette

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Contents

References
[1] Alspach, B. 1989. Lifting Hamilton cycles of quotient graphs. Discrete Math., 78, 25–36.
[2] Arvind, V. 2013. The parameterized complexity of fixpoint free elements and bases in permutation groups. Pages 4–15 of: Gutin, G., and Szeider, S. (eds), Parameterized and Exact Computation. Lecture Notes in Computer Science, vol. 8246. Springer, Switzerland.
[3] Aschbacher, M. 1984. On the maximal subgroups of the finite classical groups. Invent. Math., 76, 469–514.
[4] Aschbacher, M. 2000. Finite Group Theory (Second edition). Cambridge Studies in Advanced Mathematics, vol. 10. Cambridge University Press, Cambridge.
[5] Aschbacher, M., and Seitz, G. M. 1976. Involutions in Chevalley groups over fields of even order. Nagoya Math. J., 63, 1–91.
[6] Bamberg, J., Giudici, M., Liebeck, M.W., Praeger, C. E., and Saxl, J. 2013. The classification of almost simple 3/2 -transitive groups. Trans. Am. Math. Soc., 365, 4257–4311.
[7] Bang, A. S. 1886. Taltheoretiske undersølgelser. Tidskrifft Math., 5, 70–80, 130–137.
[8] Bereczky, Á. 1995. Fixed-point-free p-elements in transitive permutation groups. Bull. London Math. Soc., 27, 447–452.
[9] Bienert, R., and Klopsch, B. 2010. Automorphism groups of cyclic codes. J. Algebraic Combin., 31, 33–52.
[10] Biggs, N. 1973. Three remarkable graphs. Can. J. Math., 25, 397–411.
[11] Bosma, W., Cannon, J., and Playoust, C. 1997. The Magma algebra system I: The user language. J. Symbolic Comput., 24, 235–265.
[12] Boston, N., Dabrowski, W., Foguel, T., Gies, P. J., Jackson, D. A., Leavitt, J., and Ose, D. T. 1993. The proportion of fixed-point-free elements of a transitive permutation group. Commun. Algebra, 21, 3259–3275.
[13] Bray, J. N., Holt, D. F., and Roney-Dougal, C.M. 2013. TheMaximal Subgroups of the Low-Dimensional Finite Classical Groups. London Mathematical Society Lecture Note Series, vol. 407. Cambridge University Press, Cambridge.
[14] Breuer, T., Guralnick, R. M., and Kantor, W. M. 2008. Probabilistic generation of finite simple groups, II. J. Algebra, 320, 443–494.
[15] Britnell, J. R., and Maróti, A. 2013. Normal coverings of linear groups. Algebra Number Theory, 7, 2085–2102.
[16] Bubboloni, D., Praeger, C. E., and Spiga, P. 2013. Normal coverings and pairwise generation of finite alternating and symmetric groups. J. Algebra, 390, 199–215.
[17] Burness, T. C. 2007a. Fixed point ratios in actions of finite classical groups, I. J. Algebra, 309, 69–79.
[18] Burness, T. C. 2007b. Fixed point ratios in actions of finite classical groups, II. J. Algebra, 309, 80–138.
[19] Burness, T. C. 2007c. Fixed point ratios in actions of finite classical groups, III. J. Algebra, 314, 693–748.
[20] Burness, T. C. 2007d. Fixed point ratios in actions of finite classical groups, IV. J. Algebra, 314, 749–788.
[21] Burness, T. C. 2007e. On base sizes for actions of finite classical groups. J. London Math. Soc., 75, 545–562.
[22] Burness, T. C., and Giudici, M. On 2'-elusive biquasiprimitive permutation groups. In preparation.
[23] Burness, T. C., Giudici, M., and Wilson, R. A. 2011b. Prime order derangements in primitive permutation groups. J. Algebra, 341, 158–178.
[24] Burness, T. C., and Guest, S. 2013. On the uniform spread of almost simple linear groups. Nagoya Math. J., 209, 35–109.
[25] Burness, T. C., Guralnick, R.M., and Saxl, J. 2011a. On base sizes for symmetric groups. Bull. London Math. Soc., 43, 386–391.
[26] Burness, T. C., Guralnick, R. M., and Saxl, J. 2014. Base sizes for S-actions of finite classical groups. Isr. J. Math., 199, 711–756.
[27] Burness, T. C., Liebeck, M. W., and Shalev, A. 2009. Base sizes for simple groups and a conjecture of Cameron. Proc. London Math. Soc., 98, 116–162.
[28] Burness, T. C., O'Brien, E. A., and Wilson, R. A. 2010. Base sizes for sporadic simple groups. Isr. J. Math., 177, 307–333.
[29] Burness, T. C., Praeger, C. E., and Seress, Á. 2012a. Extremely primitive classical groups. J. Pure Appl. Algebra, 216, 1580–1610.
[30] Burness, T. C., Praeger, C. E., and Seress, Á. 2012b. Extremely primitive sporadic and alternating groups. Bull. London Math. Soc., 44, 1147–1154.
[31] Burness, T. C., and Tong-Viet, H. P. 2015. Derangements in primitive permutation groups, with an application to character theory. Q. J. Math., 66, 63–96.
[32] Burness, T. C., and Tong-Viet, H. P. Primitive permutation groups and derangements of prime power order. Manuscripta Math. In press.
[33] Burnside, W. 1911. Theory of Groups of Finite Order (Second edition). Cambridge University Press, Cambridge.
[34] Cameron, P. J. (ed.). 1997. Research problems from the Fifteenth British Combinatorial Conference (Stirling, 1995). Discrete Math., 167/168, 605–615.
[35] Cameron, P. J. 1999. Permutation Groups. London Mathematical Society Student Texts, vol. 45. Cambridge University Press, Cambridge.
[36] Cameron, P. J. 2000. Notes on Classical Groups. Unpublished lecture notes, available at www.maths.qmul.ac.uk/~pjc/class_gps/cg.pdf.
[37] Cameron, P. J., and Cohen, A. M. 1992. On the number of fixed point free elements in a permutation group. Discrete Math., 106/107, 135–138.
[38] Cameron, P. J., Frankl, P., and Kantor, W. M. 1989. Intersecting families of finite sets and fixed-point-free 2-elements. Eur. J. Combin., 10, 149–160.
[39] Cameron, P. J., Giudici, M., Jones, G. A., Kantor, W. M., Klin, M. H., Marušič, D., and Nowitz, L. A. 2002. Transitive permutation groups without semiregular subgroups. J. London Math. Soc., 66, 325–333.
[40] Carter, R. W. 1989. Simple Groups of Lie Type. Wiley Classics Library. John Wiley & Sons, New York.
[41] Conway, J. H., Curtis, R. T., Norton, S. P., Parker, R. A., and Wilson, R. A. 1985. Atlas of Finite Groups. Oxford University Press, Eynsham.
[42] Crestani, E., and Lucchini, A. 2012. Normal coverings of solvable groups. Arch. Math. (Basel), 98, 13–18.
[43] Crestani, E., and Spiga, P. 2010. Fixed-point-free elements in p-groups. Isr. J. Math., 180, 413–424.
[44] Diaconis, P., Fulman, J., and Guralnick, R. 2008. On fixed points of permutations. J. Algebraic Combin., 28, 189–218.
[45] Dickson, L. E. 1901. Linear Groups, with an Exposition of the Galois Field Theory. B. G. Teubner, Leipzig.
[46] Dieudonné, J. 1951. On the automorphisms of the classical groups. Mem. Am. Math. Soc., 2.
[47] Dieudonné, J. 1955. La Géométrie des Groupes Classiques. Springer, Berlin.
[48] Dixon, J. D., and Mortimer, B. 1996. Permutation Groups. Graduate Texts in Mathematics, vol. 163. Springer, New York.
[49] Dobson, E., Malnič, A., Marušič, D., and Nowitz, L. A. 2007a. Minimal normal subgroups of transitive permutation groups of square-free degree. Discrete Math., 307, 373–385.
[50] Dobson, E., Malnič, A., Marušič, D., and Nowitz, L. A. 2007b. Semiregular automorphisms of vertex-transitive graphs of certain valencies. J. Combin. Theory Ser. B, 97, 371–380.
[51] Dobson, E., and Marušič, D. 2011. On semiregular elements of solvable groups. Commun. Algebra, 39, 1413–1426.
[52] Fein, B., Kantor, W. M., and Schacher, M. 1981. Relative Brauer groups, II. J. Reine Angew. Math., 328, 39–57.
[53] Feit, W. 1980. Some consequences of the classification of finite simple groups. Pages 175–181 of: The Santa Cruz Conference on Finite Groups, 1979. Proceeding of Symposia in Pure Mathematics, vol. 37. American Mathematical Society, Providence, RI.
[54] Frucht, R. 1970. How to describe a graph. Ann. N. Y. Acad. Sci., 175, 159–167.
[55] Fulman, J., and Guralnick, R. M. 2003. Derangements in simple and primitive groups. Pages 99–121 of: Groups, Combinatorics & Geometry (Durham, 2001). World Scientific, River Edge, NJ.
[56] Fulman, J., and Guralnick, R. M. 2012. Bounds on the number and sizes of conjugacy classes in finite Chevalley groups with applications to derangements. Trans. Am. Math. Soc., 364, 3023–3070.
[57] Fulman, J., and Guralnick, R. M. Derangements in finite classical groups for actions related to extension field and imprimitive subgroups and the solution of the Boston-Shalev conjecture. Submitted (arxiv:1508.00039).
[58] Fulman, J., and Guralnick, R. M.Derangements in subspace actions of finite classical groups. Trans. Am. Math. Soc., to appear.
[59] Galois, E. 1846. Oeuvres mathématiques: Lettre de Galois à M. Auguste Chevalier (29 Mai 1832). J. Math. Pures Appl. (Liouville), 11, 408–415.
[60] Gill, N. 2007. Polar spaces and embeddings of classical groups. N. Z. J. Math., 36, 175–184.
[61] Giudici, M. 2003. Quasiprimitive groups with no fixed point free elements of prime order. J. London Math. Soc., 67, 73–84.
[62] Giudici, M. 2007. New constructions of groups without semiregular subgroups. Commun. Algebra, 35, 2719–2730.
[63] Giudici, M., and Kelly, S. 2009. Characterizing a family of elusive permutation groups. J. Group Theory, 12, 95–105.
[64] Giudici, M., Morgan, L., Potočnik, P., and Verret, G. 2015. Elusive groups of automorphisms of digraphs of small valency. Eur. J. Combin., 46, 1–9.
[65] Giudici, M., and Xu, J. 2007. All vertex-transitive locally-quasiprimitive graphs have a semiregular automorphism. J. Algebraic Combin., 25, 217–232.
[66] Gorenstein, D., and Lyons, R. 1983. The local structure of finite groups of characteristic 2 type. Mem. Am. Math. Soc., 276.
[67] Gorenstein, D., Lyons, R., and Solomon, R. 1998. The Classification of the Finite Simple Groups. Number 3. Mathematical Surveys and Monographs, vol. 40. American Mathematical Society, Providence, RI.
[68] Guralnick, R. M. 1990. Zeroes of permutation characters with applications to prime splitting and Brauer groups. J. Algebra, 131, 294–302.
[69] Guralnick, R. M.Conjugacy classes of derangements in finite transitive groups. Proc. Steklov Inst. Math. In press.
[70] Guralnick, R. M., and Kantor, W. M. 2000. Probabilistic generation of finite simple groups. J. Algebra, 234, 743–792.
[71] Guralnick, R. M., Müller, P., and Saxl, J. 2003. The rational function analogue of a question of Schur and exceptionality of permutation representations. Mem. Am. Math. Soc., 773.
[72] Guralnick, R. M., and Saxl, J. 2003. Generation of finite almost simple groups by conjugates. J. Algebra, 268, 519–571.
[73] Guralnick, R. M., and Wan, D. 1997. Bounds for fixed point free elements in a transitive group and applications to curves over finite fields. Isr. J. Math., 101, 255–287.
[74] Hartley, R. W. 1925. Determination of the ternary collineation groups whose coefficients lie in the GF(2n). Ann. Math., 27, 140–158.
[75] Herstein, I. N. 1975. Topics in Algebra (Second edition). John Wiley & Sons, New York.
[76] Hiss, G., and Malle, G. 2001. Low-dimensional representations of quasi-simple groups. LMS J. Comput. Math., 4, 22–63.
[77] Isbell, J. R. 1957. Homogeneous games. Math. Student, 25, 123–128.
[78] Isbell, J. R. 1960. Homogeneous games. II. Proc. Am. Math. Soc., 11, 159–161.
[79] Isbell, J. R. 1964. Homogeneous games. III. Pages 255–265 of: Advances in Game Theory. Princeton University Press, Princeton, NJ.
[80] Jones, G. A. 2002. Cyclic regular subgroups of primitive permutation groups. J. Group Theory, 5, 403–407.
[81] Jones, J. W., and Roberts, D. P. 2014. The tame-wild principle for discriminant relations for number fields. Algebra Number Theory, 8, 609–645.
[82] Jordan, C. 1872. Recherches sur les substitutions. J. Math. Pures Appl. (Liouville), 17, 351–367.
[83] Jordan, D. 1988. Eine Symmetrieeigenschaft von Graphen. Pages 17–20 of: Graphentheorie und ihre Anwendungen (Stadt Wehlen, 1988). Dresdner Reihe Forsch., vol. 9. Päd. Hochsch., Dresden.
[84] Khukhro, E. I., and Mazurov, V. D. (eds.). 2014. The Kourovka Notebook: Unsolved Problems in Group Theory (Eighteenth edition). Institute of Mathematics, Novosibirsk.
[85] Kleidman, P. B. 1987. The maximal subgroups of the finite 8-dimensional orthogonal groups PO8+ (q) and of their automorphism groups. J. Algebra, 110, 173–242.
[86] Kleidman, P., and Liebeck, M. 1990. The Subgroup Structure of the Finite Classical Groups. London Mathematical Society Lecture Note Series, vol. 129. Cambridge University Press, Cambridge.
[87] Knapp, A.W. 2007. Advanced Algebra. Cornerstones. Birkhäuser, Boston, MA.
[88] Kutnar, K., and Šparl, P. 2010. Distance-transitive graphs admit semiregular automorphisms. Eur. J. Combin., 31, 25–28.
[89] Leighton, F. T. 1983. On the decomposition of vertex-transitive graphs into multicycles. J. Res. Natl. Bur. Stand., 88, 403–410.
[90] Li, C. H. 2003. The finite primitive permutation groups containing an abelian regular subgroup. Proc. London Math. Soc., 87, 725–747.
[91] Li, C. H. 2005. Permutation groups with a cyclic regular subgroup and arc transitive circulants. J. Algebraic Combin., 21, 131–136.
[92] Li, C. H. 2006. Finite edge-transitive Cayley graphs and rotary Cayley maps. Trans. Am. Math. Soc., 358, 4605–4635.
[93] Liebeck, M. W., and O'Brien, E. A. Conjugacy classes in finite groups of Lie type: representatives, centralizers and algorithms. In preparation.
[94] Liebeck, M. W., Praeger, C. E., and Saxl, J. 1988. On the O'Nan–Scott theorem for finite primitive permutation groups. J. Aust. Math. Soc. Ser. A, 44, 389–396.
[95] Liebeck, M. W., Praeger, C. E., and Saxl, J. 2000. Transitive subgroups of primitive permutation groups. J. Algebra, 234, 291–361.
[96] Liebeck, M. W., and Seitz, G. M. 2012. Unipotent and Nilpotent Classes in Simple Algebraic Groups and Lie Algebras. Mathematical Surveys and Monographs, vol. 180. American Mathematical Society, Providence, RI.
[97] Liebeck, M. W., and Shalev, A. 1999. Simple groups, permutation groups, and probability. J. Am. Math. Soc., 12, 497–520.
[98] Lübeck, F. 2001. Small degree representations of finite Chevalley groups in defining characteristic. LMS J. Comput. Math., 4, 135–169.
[99] Malle, G., and Testerman, D. 2011. Linear Algebraic Groups and Finite Groups of Lie Type. Cambridge Studies in Advanced Mathematics, vol. 133. Cambridge University Press, Cambridge.
[100] Malnič, A., Marušič, D., Šparl, P., and Frelih, B. 2007. Symmetry structure of bicirculants. Discrete Math., 307, 409–414.
[101] Marušič, D. 1981. On vertex symmetric digraphs. Discrete Math., 36, 69–81.
[102] Marusic, D., and Scapellato, R. 1998. Permutation groups, vertex-transitive digraphs and semiregular automorphisms. Eur. J. Combin., 19, 707–712.
[103] McKay, B. D., and Royle, G. F. 1990. The transitive graphs with at most 26 vertices. Ars Combin., 30, 161–176.
[104] Mitchell, H. H. 1911. Determination of the ordinary and modular ternary linear groups. Trans. Am. Math. Soc., 12, 207–242.
[105] Mitchell, H. H. 1914. The subgroups of the quaternary abelian linear group. Trans. Am. Math. Soc., 15, 379–396.
[106] Montmort, P. R. de. 1708. Essay d'analyse sur les Jeux de Hazard. Quillau, Paris.
[107] Neumann, P. M., and Praeger, C. E. 1998. Derangements and eigenvalue-free elements in finite classical groups. J. London Math. Soc., 58, 564–586.
[108] Pless, V. 1964. On Witt's theorem for nonalternating symmetric bilinear forms over a field of characteristic 2. Proc. Am. Math. Soc., 15, 979–983.
[109] Praeger, C. E., Li, C. H., and Niemeyer, A. C. 1997. Finite transitive permutation groups and finite vertex-transitive graphs. Pages 277–318 of: Graph Symmetry (Montreal, 1996). NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 497. Kluwer, Dordrecht.
[110] Sabidussi, G. 1958. On a class of fixed-point-free graphs. Proc. Am. Math. Soc., 9, 800–804.
[111] Saxl, J., and Seitz, G. M. 1997. Subgroups of algebraic groups containing regular unipotent elements. J. London Math. Soc., 55, 370–386.
[112] Schur, I. 1933. Zur Theorie der einfach transitiven Permutationsgruppen. S. B. Preuss. Akad. Wiss., Phys.-Math. Kl., 598–623.
[113] Serre, J.-P. 2003. On a theorem of Jordan. Bull. Am. Math. Soc., 40, 429–440.
[114] Spiga, P. 2013. Permutation 3-groups with no fixed-point-free elements. Algebra Colloq., 20, 383–394.
[115] Steinberg, R. 1968. Lectures on Chevalley Groups. Department of Mathematics, Yale University.
[116] Suzuki, M. 1986. Group Theory II. Springer, New York.
[117] Takács, L. 1979/1980. The problem of coincidences. Arch. Hist. Exact Sci., 21, 229–244.
[118] Taylor, D. E. 1992. The Geometry of the Classical Groups. Sigma Series in Pure Mathematics, vol. 9. Heldermann Verlag, Berlin.
[119] Wall, G. E. 1963. On the conjugacy classes in the unitary, symplectic and orthogonal groups. J. Aust. Math. Soc., 3, 1–62.
[120] Wielandt, H. 1964. Finite Permutation Groups. Academic Press, New York.
[121] Wilson, R. A. 1985. Maximal subgroups of automorphism groups of simple groups. J. London Math. Soc., 32, 460–466.
[122] Wilson, R. A. 2009. The Finite Simple Groups. Graduate Texts in Mathematics, vol. 251. Springer, London.
[123] Zsigmondy, K. 1892. Zur Theorie der Potenzreste. Monatsh. Math. Phys., 3, 265–284.

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