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  3. The Maximal Subgroups of the Low-Dimensional Finite Classical Groups
  • Cited by 196
Publisher:
Cambridge University Press
Online publication date:
July 2013
Print publication year:
2013
Online ISBN:
9781139192576
DOI:
https://doi.org/10.1017/CBO9781139192576
Subjects:
Mathematics (general), Mathematics, Algebra
Series:
London Mathematical Society Lecture Note Series (407)
Information

Book description

This book classifies the maximal subgroups of the almost simple finite classical groups in dimension up to 12; it also describes the maximal subgroups of the almost simple finite exceptional groups with socle one of Sz(q), G2(q), 2G2(q) or 3D4(q). Theoretical and computational tools are used throughout, with downloadable Magma code provided. The exposition contains a wealth of information on the structure and action of the geometric subgroups of classical groups, but the reader will also encounter methods for analysing the structure and maximality of almost simple subgroups of almost simple groups. Additionally, this book contains detailed information on using Magma to calculate with representations over number fields and finite fields. Featured within are previously unseen results and over 80 tables describing the maximal subgroups, making this volume an essential reference for researchers. It also functions as a graduate-level textbook on finite simple groups, computational group theory and representation theory.

Reviews

'The study of the maximal subgroups of the finite classical groups started long ago. Their understanding has been crucial in the work on the classification of finite simple groups. The methods [in this book] efficiently combine theoretical techniques together with computer algebra systems (GAP, Magma), with the computational files available [online] … The theoretical literature on the subject is highly exploited, especially Aschbacher's results which describe the maximal subgroups of most of the finite almost simple classical groups … The tables are packed with a great amount of information, and should satisfy the reader who is looking for some specific information about the structure of the maximal subgroups of a given classical group.'

Nadia P. Mazza Source: Mathematical Reviews

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Contents

Contents
References
References
[1] M., Aschbacher. On the maximal subgroups of the finite classical groups. Invent. Math. 76 (<>1984), 469–514.
[2] M., Aschbacher. Chevalley groups of type G2 as the group of a trilinear form. J. Algebra 109 (1987), 193–259.
[3] D.M., Bloom. The subgroups of PSL(3, q)for odd q. Trans. Amer. Math. Soc. 127 (1967), 150–178.
[4] R., Brauer and C., Nesbitt. On the modular characters of groups. Ann. of Math. (2) 42 (1941), 556–590.
[5] W., Bosma, J., Cannon and C., Playoust. The Magma algebra system. I. The user language. J. Symbolic Comput. 24 (1997), 235–265.
[6] J. N., Bray, D. F., Holt and C.M., Roney-Dougal. Certain classical groups are not well-defined. J. Group Theory 12 (2009), 171–180.
[7] J., Brundan and A., Kleschev. Lower bounds for the degrees of representations of irreducible Brauer characters of finite general linear groups. J. Algebra 223 (2000), 615–629.
[8] Peter J., Cameron. Permutation Groups. London Math. Soc. Student Texts, 45. Cambridge University Press, Cambridge, 1999.
[9] J. J., Cannon and D.F., Holt. Computing maximal subgroups of inite groups. J. Symbolic Comput. 37 (<>2004), 589–609.
[10] Roger W., Carter. Simple Groups of Lie Type. John Wiley and Sons, London-New York-Sydney, 1972.
[11] P.M., Cohn. Basic Algebra: Groups, Rings and Fields. Springer-Verlag, London, 2003.
[12] J. H., Conway, R. T., Curtis, S. P., Norton, R.A., Parker and R.A., Wilson. An ATLAS of Finite Groups. Clarendon Press, Oxford, 1985; reprinted with corrections 2003.
[13] B. N., Cooperstein. The geometry of root subgroups in exceptional groups. I. Geom. Dedicata 8 (1979), 317–338.
[14] B.N., Cooperstein. Maximal subgroups of G2(2n). J. Algebra 70 (1981), 23–36.
[15] A., Cossidente and O.H., King. Maximal subgroups of inite orthogonal groups stabilizing spreads of lines. Comm. Algebra 34 (2006), 4291–4309.
[16] A., Cossidente and O.H., King. On twisted tensor product group embeddings and the spin representation of symplectic groups. Adv. Geom. 7 (<>2007), 55–64.
[17] A., Cossidente and A., Siciliano. On some maximal subgroups in Aschbacher's class c5. Linear Algebra Appl. 403 (2005), 285–290.
[18] H.J., Coutts, M. R., Quick and C.M., Roney-Dougal. The primitive groups of degree less than 4096. Comm. Algebra 39 (2011), 3526–3546.
[19] Charles, W.Curtis and Irving Reiner. Representation Theory of Finite Groups and Associative Algebras. Reprint of the 1962 original. Wiley Classics Library. John Wiley & Sons, Inc., New York, 1988.
[20] Charles, W.Curtis and Irving Reiner. Methods of Representation Theory. Vol. II. With applications to inite groups and orders. John Wiley & Sons, Inc., New York, 1987.
[21] M. R., Darafsheh. Maximal subgroups of the group GL6(2). Bull. Malaysian Math.Soc. (2) 7 (1984), 49–55.
[22] Leonard Eugene, Dickson. Linear groups, with an exposition of the Galois field theory. Teubner, Leipzig, 1901 (Dover reprint 1958).
[23] L., Di Martino and A., Wagner. The irreducible subgroups of PSL(V5, q), where q is odd. Resultate Math. 2 (1979), 54–61.
[24] L., Finkelstein and A., Rudvalis. Maximal subgroups of the Hall-Janko-Wales group. J. Algebra 24 (1973), 486–493.
[25] L., Finkelstein and A., Rudvalis. The maximal subgroups of Janko's simple group of order 50, 232, 960. J. Algebra 30 (1974), 122–143.
[26] D. E., Flesner. Finite symplectic geometry in dimension four and characteristic two. Illinois J. Math. 19 (1975), 41–47.
[27] D. E., Flesner. Maximal subgroups of PSp4(2n) containing central elations or noncentered skew elations. Illinois J. Math. 19 (1975), 247–268.
[28] William, Fulton and Joe, Harris. Representation Theory. Graduate Texts in Mathematics 129, Springer-Verlag, New York, 1991.
[29] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.5.7; 2012. <http://www.gap-system.org>
[30] Nick, Gill. Polar spaces and embeddings of classical groups. New Zealand J. Math. 36 (2007), 175–184.
[31] Daniel, Gorenstein. Finite Groups. Harper and Row, New York–London, 1968.
[32] Daniel, Gorenstein. The classiication of inite simple groups, Number 3, Part I, Chapter A, Almost simple K-groups. Mathematical Surveys and Monographs, 40.3. American Mathematical Society, Providence, RI, 1998.
[33] Ronald L., Graham, Donald E., Knuth and Oren, Patashnik. Concrete mathematics. A foundation for computer science. Second edition. Addison-Wesley Publishing Company, Reading, MA, 1994.
[34] R. M., Guralnick, K., Magaard, J., Saxl and P.H., Tiep. Cross characteristic representations of symplectic and unitary groups. J. Algebra 257 (2002), 291–347. Addendum: J. Algebra 299 (2006), 443–446.
[35] Robert, Guralnick and Gunter, Malle. Products of conjugacy classes and fixed point spaces. J. Amer. Math. Soc. 25 (2012), 77–121.
[36] R., Guralnick, T., Penttila, C. E., Praeger and J., Saxl. Linear groups with orders having certain large prime divisors. Proc. London Math. Soc. 78 (1999), 167–214.
[37] R. M., Guralnick and P.H., Tiep. Low-dimensional representations of special linear groups in cross characteristics. Proc. London Math. Soc. 78 (1999), 116–138.
[38] R. M., Guralnick and P.H., Tiep. Cross characteristic representations of even characteristic symplectic groupsTrans. Amer. Math. Soc. 356 (2004), 4969–5023.
[39] K., Harada and H., Yamaki. The irreducible subgroups of GLn(2) with n ≤ 6. C. R. Math. Rep. Acad. Sci. Canada, 1 (1978/1979), 75–78.
[40] R. W., Hartley. Determination of the ternary collineation groups whose coefficients lie in the GF(2n). Ann. of Math. 27 (1925/1926), 140–158.
[41] G., Hiß, W. J., Husen and K., Magaard. Imprimitive irreducible modules for finite quasisimple groups. <http://arxiv.org/abs/1211.6350>.
[42] G., Hiß and G., Malle. Low-dimensional representations of quasi-simple groups. LMS J. Comput. Math. 4 (2001), 22–63. Corrigenda: LMS J. Comput. Math. 5 (2002), 95–126.
[43] G., Hiß and G., Malle. Low-dimensional representations of special unitary groups. J. Algebra 236 (2001), 747–767.
[44] D. F., Holt and S., Rees. Testing modules for irreducibility. J. Austral. Math. Soc. Ser. A 57 (1994), 1–16.
[45] D. F., Holt and C.M., Roney-Dougal. Constructing maximal subgroups of classical groups. LMS J. Comput. Math. 8 (2005), 46–79.
[46] D. F., Holt and C.M., Roney-Dougal. Constructing maximal subgroups of orthogonal groups. LMS J. Comput. Math. 13 (2010), 164–191.
[47] C., Hoffman. Cross characteristic projective representations for some classical groups. J. Algebra 229 (2000), 666–677.
[48] C.H., Houghton. Wreath products of groupoids. J. London Math. Soc. (2) 10 (1975), 179–188.
[49] R. B., Howlett, L. J., Rylands and D.E., Taylor. Matrix generators for exceptional groups of Lie type. J. Symbolic Comput. 31 (2001), 429–445.
[50] James E., Humphreys. Linear Algebraic Groups. Graduate Texts in Mathematics, 21. Springer-Verlag, Berlin, 1975.
[51] J. E., Humphreys. Modular representations of finite groups of Lie type. In Finite Simple Groups II, Durham, 1978. Ed. M. J., Collins, Academic Press, London (1980), 259–290.
[52] B., Huppert. Singer-Zyklen in klassischen Gruppen. Math. Z. 117 (1970), 141–150.
[53] B., Huppert. Endliche Gruppen I. Springer-Verlag, Berlin, 1967.
[54] W. J., Husen. Maximal embeddings of alternating groups in the classical groups. Ph.D. Thesis, Wayne State University, 1997.
[55] W. J., Husen. Irreducible modules for classical and alternating groups. J. Algebra 226 (2000), 977–989.
[56] I., Martin Isaacs. Character theory of finite groups. Pure Appl. Math., 69. Academic Press, New York-London, 1976.
[57] Christoph, Jansen, Klaus, Lux, Richard, Parker and Robert, Wilson. An Atlas of Brauer characters. The Clarendon Press, Oxford University Press, New York, 1995.
[58] Jens Carsten, Jantzen. Representations of Algebraic Groups. Pure Appl. Math., 131. Academic Press, Boston MA, 1987.
[59] W. M., Kantor and R.A., Liebler. The rank 3 permutation representations of the finite classical groups. Trans. Amer. Math. Soc. 271 (1982), 1–71.
[60] O. H., King. The subgroup structure of inite classical groups in terms of geometric configurations. In Surveys in combinatorics, 2005. Ed. B. S., Webb, London Math. Soc. Lecture Note Ser., 327. Cambridge University Press, Cambridge, 2005, 29–56
[61] P. B., Kleidman. The maximal subgroups of the low-dimensional classical groups. PhD Thesis, University of Cambridge, 1987.
[62] P. B., Kleidman. The maximal subgroups of the inite 8-dimensional orthogonal groups PΩ+8 (q) and of their automorphism groups. J. Algebra 110 (1987), 173–242.
[63] P. B., Kleidman. The maximal subgroups of the Steinberg triality groups 3D4(q) and of their automorphism groups. J. Algebra 115 (1988), 182–199.
[64] P. B., Kleidman. The maximal subgroups of the Chevalley groups G2(q) with q odd, the Ree groups 2G2(q), and their automorphism groups. J. Algebra 117 (1988), 30–71.
[65] P. B., Kleidman and M.W., Liebeck. A survey of the maximal subgroups of the inite simple groups. Geom. Dedicata 25 (1988), 375–389.
[66] Peter, Kleidman and Martin, Liebeck. The subgroup structure of the inite classical groups. London Math. Soc. Lecture Note Ser., 129. Cambridge University Press, Cambridge, 1990.
[67] A. S., Kondrat'ev. Irreducible subgroups of the group GL(7, 2). Mat. Zametki 37 (1985), 317–321, 460.
[68] A. S., Kondrat'ev. Linear groups of small degree over a field of order 2 (Russian). Algebra i Logika 25 (1986), 544–565.
[69] A. S., Kondrat'ev. Irreducible subgroups of the group GL(9, 2). Mat. Zametki 39 (1986), 320–329, 460.
[70] A. S., Kondrat'ev. The irreducible subgroups of the group GL8(2). Comm. Algebra 15 (1987), 1039–1093.
[71] A. S., Kondrat'ev. Finite linear groups of degree 6. Algebra i Logika 28 (1989), 181–206, 245.
[72] A. S., Kondratiev. Finite linear groups of small degree. In The Atlas of Finite Groups: Ten Years On (Birmingham, 1995). London Math. Soc. Lecture Note Ser., 249. Cambridge University Press, Cambridge, 1990, 139–148.
[73] A. S., Kondrat'ev. Finite linear groups of small degree. II. Comm. Algebra 29 (2001), 4103–4123.
[74] Serge, Lang. Algebraic Number Theory. Addison-Wesley, Reading, Mass.–London–Don Mills, Ont., 1970.
[75] V., Landazuri and G.M., Seitz. On the minimal degrees of projective representations of the inite Chevalley groups. J. Algebra 32 (1974), 418–443.
[76] V. M., Levchuk and Ya. N., Nuzhin. Structure of Ree groups. Algebra i Logika 24 (1985), 26–41.
[77] M. W., Liebeck. On the orders of maximal subgroups of the inite classical groups. Proc. London Math. Soc. (3) 50 (1985), 426–446.
[78] M. W., Liebeck, C. E., Praeger and J., Saxl. A classiication of the maximal subgroups ofthe inite alternating and symmetric groups. J. Algebra 111 (1987), 365–383.
[79] M. W., Liebeck, C. E., Praeger and J., Saxl. On the O'Nan–Scott theorem for inite primitive permutation groups. J. Austral. Math. Soc. Ser. A 44 (1988), 389–396.
[80] M. W., Liebeck and J., Saxl. Primitive permutation groups containing an element of large prime order. J. London Math. Soc. 31 (1985), 237–249.
[81] M. W., Liebeck, J., Saxl and G.M., Seitz. On the overgroups of irreducible subgroups of the finite classical groups. Proc. London Math. Soc. 55 (1987), 507–537.
[82] M. W., Liebeck and G.M., Seitz. On the subgroup structure of classical groups. Invent. Math. 134 (1998), 427–453.
[83] M. W., Liebeck and G.M., Seitz. A survey of maximal subgroups of exceptional groups of Lie type. In Groups, combinatorics & geometry (Durham, 2001). World Sci. Publ., River Edge, NJ, (2003), 139–146.
[84] F., Lübeck. Small degree representations of finite Chevalley groups in defining characteristic. LMS J. Comput. Math. 4 (2001), 135–169.
[85] F., Lübeck. Tables of Weight Multiplicities, <http://www.math.rwth-aachen.de/~Frank.Luebeck/chev/WMSmall/>
[86] F., Lübeck. Conway polynomials for finite fields. <http://www.math.rwth-aachen.de/~Frank.Luebeck/data/ConwayPol/>
[87] Klaus, Lux and Herbert, Pahlings. Representations of groups: a computational approach. Cambridge studies in advanced mathematics, 124. Cambridge University Press, Cambridge, 2010.
[88] K., Magaard, G., Roehrle and D., Testerman. On the irreducibility of sym-metrizations of cross-characteristic representations of finite classical groups. <http://arxiv.org/abs/1201.2057>.
[89] K., Magaard, G., Malle and P.H., Tiep. Irreducibility of tensor squares, symmetric squares and alternating squares. Pacific J. Math. 202 (2002), 379–427.
[90] G., Malle. The maximal subgroups of 2F4(q2). J. Algebra 139 (1991), 52–69.
[91] Gunter, Malle and Donna, Testerman. Linear algebraic groups and finite groups of Lie type. Cambridge studies in advanced mathematics, 133. Cambridge University Press, Cambridge, 2011.
[92] H. H., Mitchell. Determination of the ordinary and modular ternary linear groups. Trans. Amer. Math. Soc. 12 (1911), 207–242.
[93] H. H., Mitchell. The subgroups of the quaternary abelian linear group. Trans. Amer. Math. Soc. 15 (1914), 379–396.
[94] E. H., Moore. The subgroups of the generalized finite modular group. Dicennial publications of the University of Chicago 9 (1904), 141–190.
[95] B., Mwene. On the subgroups of the group PSL4(2m). J. Algebra 41 (1976), 79–107.
[96] B., Mwene. On some subgroups of the group PSL4(q), q odd. Geom. Dedicata 12 (1982), 189–199.
[97] W., Nickel. Endliche Körper in dem gruppentheoretischen Programmsystem GAP. Diplomarbeit, RWTH, Aachen, 1988.
[98] C. M., Roney-Dougal. The primitive groups of degree less than 2500. J. Algebra 292 (2005), 154–183.
[99] M., Schaffer. Twisted tensor product subgroups of finite classical groups. Comm. Algebra 27 (1999), 5097–5166.
[100] Gary M., Seitz. The maximal subgroups of classical algebraic groups. Mem. Amer. Math. Soc. 67 (1987).
[101] G. M., Seitz and A.E., Zalesskii. On the minimal degrees of projective representations of the finite Chevalley groups. II. J. Algebra 158 (1993), 233–243.
[102] C.C., Sims. Computational methods in the study of permutation groups. In Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967). Pergamon, Oxford (1970), 169–183.
[103] A. K., Steel. Construction of ordinary irreducible representations of finite groups. PhD Thesis, University of Sydney, 2012.
[104] R., Steinberg. Representations of algebraic groups. Nagoya Math. J. 22 (1963), 33–56.
[105] Robert, Steinberg. Lectures on Chevalley Groups. Yale University Mathematics Department, 1968.
[106] M., Suzuki. On a class of doubly transitive groups. Ann. of Math. (2) 75 (1962), 105–145.
[107] J., Tits. Ovoïdes et groupes de Suzuki. Arch. Math. 13 (1962), 187–198.
[108] Donald E., Taylor. The geometry of the classical groups. Heldermann Verlag, Berlin, 1992.
[109] P. H., Tiep and A.E., Zalesskii. Some aspects of finite linear groups: a survey. J. Math. Sci. (New York) 100 (2000), 1893–1914.
[110] A., Wagner. The subgroups of PSL(5, 2a). Resultate Math. 1 (1978), 207–226.
[111] R. A., Wilson, P. G., Walsh, J., Tripp, I. A. I., Suleiman, R. A., Parker, S. P., Norton, S. J., Nickerson, S. A., Linton, J. N., Bray and R.A., Abbott. Atlas of Finite Group Representations. <http://brauer.maths.qmul.ac.uk/Atlas/v3/>
[112] R. A., Wilson. The complex Leech lattice and maximal subgroups of the Suzuki groups. J. Algebra 84 (1983), 151–188.
[113] R. A., Wilson. Maximal subgroups of automorphism groups of simple groups. J. London Math. Soc. (2) 32 (1985), 460–466.
[114] Robert A., Wilson. The Finite Simple Groups. Graduate Texts in Mathematics, 251. Springer-Verlag London, Ltd., London, 2009.
[115] A., Wiman. Bestimmung aller Untergruppen einer doppelt unendlichen Reihe von einfachen Gruppen. Stockh. Akad. Bihang 25 (1899), 1–47.
[116] A. E., Zalesskii. Classification of the finite linear groups of degrees 4 and 5 over a field of characteristic 2. Dokl. Akad. Nauk BSSR 21 (1977), 389–392, 475.
[117] A. E., Zalesskii and I.D., Suprunenko. Classification of finite irreducible linear groups of rank 4 over a field of characteristic p > 5. Vestsi Akad. Navuk BSSR Ser. Fz.-Mat. Navuk 138 (1978), 9–15.
[118] K., Zsigmondy. Zur Theorie der Potenzreste. Monatsh. für Math. u. Phys. 3 (1892), 265–284.

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