×

zbMATH — the first resource for mathematics

Products of conjugacy classes and fixed point spaces. (English) Zbl 1286.20007
This long paper contains a number of interesting results on the objects of its title. The first main result is the following.
Theorem: Let \(G\) be a finite nonabelian simple group. There exists a conjugacy class \(C\) of \(G\) such that: (i) there is a triple of elements in \(C\) which have product \(1\) and generate \(G\), and (ii) there exist \(x,y\in C\) that generate \(G\) such that \(xy\) is conjugate to \(x^2\) unless \(G\) is a projective two-dimensional special linear group \(L_2(q)\) with \(q\) even or \(q=7\).
The proof of this result is long and technical and depends on CFSG. The exceptions in (2) are true exceptions, and the authors prove that any group satisfying (2) can have no irreducible 2-dimensional representations.
As one consequence of this and earlier results [in R. Guralnick and A. Maróti, Adv. Math. 226, No. 1, 298-308 (2011; Zbl 1211.20011)] the authors are able to prove the following generalization of a conjecture of Neumann.
Theorem: Let \(G\) be a nontrivial irreducible subgroup of \(\text{GL}(V)\) with \(V\) a finite-dimensional vector space over a field \(k\). There exists an element \(g\in G\) with \(\dim C_V(g)\leq(1/3)\dim V\).
Moreover, for finite simple groups \(G\) it is proved that any element in \(G\) is the product of two \(m\)-th powers, where \(m\) a either power of \(6\) or a prime power; part of this was a conjecture of Larsen, Shalev and Tiep. The paper also contains new results on a conjecture of Thompson which states that for any finite nonabelian simple group there is a conjugacy class \(C\) such that every element of \(G\) is the product of two elements in \(C\).

MSC:
20C15 Ordinary representations and characters
20D05 Finite simple groups and their classification
20E45 Conjugacy classes for groups
20F05 Generators, relations, and presentations of groups
20G15 Linear algebraic groups over arbitrary fields
20C20 Modular representations and characters
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20E28 Maximal subgroups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
11P05 Waring’s problem and variants
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] John Bamberg and Tim Penttila, Overgroups of cyclic Sylow subgroups of linear groups, Comm. Algebra 36 (2008), no. 7, 2503 – 2543. · Zbl 1162.20033
[2] Ingrid Bauer, Fabrizio Catanese, and Fritz Grunewald, Beauville surfaces without real structures, Geometric methods in algebra and number theory, Progr. Math., vol. 235, Birkhäuser Boston, Boston, MA, 2005, pp. 1 – 42. · Zbl 1094.14508
[3] Ingrid Bauer, Fabrizio Catanese, and Fritz Grunewald, Chebycheff and Belyi polynomials, dessins d’enfants, Beauville surfaces and group theory, Mediterr. J. Math. 3 (2006), no. 2, 121 – 146. · Zbl 1167.14300
[4] Edward Bertram, Even permutations as a product of two conjugate cycles, J. Combinatorial Theory Ser. A 12 (1972), 368 – 380. · Zbl 0238.20004
[5] Thomas Breuer, Robert M. Guralnick, and William M. Kantor, Probabilistic generation of finite simple groups. II, J. Algebra 320 (2008), no. 2, 443 – 494. · Zbl 1181.20013
[6] Roger W. Carter, Finite groups of Lie type, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993. Conjugacy classes and complex characters; Reprint of the 1985 original; A Wiley-Interscience Publication.
[7] Vladimir Chernousov, Erich W. Ellers, and Nikolai Gordeev, Gauss decomposition with prescribed semisimple part: short proof, J. Algebra 229 (2000), no. 1, 314 – 332. · Zbl 0964.20024
[8] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of finite groups, Oxford University Press, Eynsham, 1985. Maximal subgroups and ordinary characters for simple groups; With computational assistance from J. G. Thackray. · Zbl 0568.20001
[9] Bruce N. Cooperstein, Maximal subgroups of \?\(_{2}\)(2\(^{n}\)), J. Algebra 70 (1981), no. 1, 23 – 36. · Zbl 0459.20007
[10] Bruce N. Cooperstein, Subgroups of exceptional groups of Lie type generated by long root elements. I. Odd characteristic, J. Algebra 70 (1981), no. 1, 270 – 282. , https://doi.org/10.1016/0021-8693(81)90259-3 Bruce N. Cooperstein, Subgroups of exceptional groups of Lie type generated by long root elements. II. Characteristic two, J. Algebra 70 (1981), no. 1, 283 – 298. , https://doi.org/10.1016/0021-8693(81)90260-X Bruce N. Cooperstein, Subgroups of exceptional groups of Lie type generated by long root elements. I. Odd characteristic, J. Algebra 70 (1981), no. 1, 270 – 282. , https://doi.org/10.1016/0021-8693(81)90259-3 Bruce N. Cooperstein, Subgroups of exceptional groups of Lie type generated by long root elements. II. Characteristic two, J. Algebra 70 (1981), no. 1, 283 – 298. · Zbl 0463.20015
[11] François Digne and Jean Michel, Representations of finite groups of Lie type, London Mathematical Society Student Texts, vol. 21, Cambridge University Press, Cambridge, 1991. · Zbl 0815.20014
[12] Erich W. Ellers and Nikolai Gordeev, On the conjectures of J. Thompson and O. Ore, Trans. Amer. Math. Soc. 350 (1998), no. 9, 3657 – 3671. · Zbl 0910.20007
[13] Walter Feit, On large Zsigmondy primes, Proc. Amer. Math. Soc. 102 (1988), no. 1, 29 – 36. · Zbl 0639.10007
[14] Paul Fong and Bhama Srinivasan, The blocks of finite general linear and unitary groups, Invent. Math. 69 (1982), no. 1, 109 – 153. · Zbl 0507.20007
[15] Paul Fong and Bhama Srinivasan, Brauer trees in classical groups, J. Algebra 131 (1990), no. 1, 179 – 225. · Zbl 0704.20011
[16] S. Garion, M. Penegini, New Beauville surfaces, moduli spaces and finite groups. Preprint, arXiv:0910.5402. · Zbl 1311.14040
[17] S. Garion, M. Larsen, A. Lubotzky, Beauville surfaces and finite simple groups. J. Reine Angew. Math., to appear. · Zbl 1255.20008
[18] Meinolf Geck, Gerhard Hiss, Frank Lübeck, Gunter Malle, and Götz Pfeiffer, CHEVIE — a system for computing and processing generic character tables, Appl. Algebra Engrg. Comm. Comput. 7 (1996), no. 3, 175 – 210. Computational methods in Lie theory (Essen, 1994). · Zbl 0847.20006
[19] J. Getz, An approach to nonsolvable base change and descent for \( \operatorname{GL}_2\) over number fields. Preprint.
[20] David Gluck, Sharper character value estimates for groups of Lie type, J. Algebra 174 (1995), no. 1, 229 – 266. · Zbl 0842.20014
[21] Rod Gow, Commutators in finite simple groups of Lie type, Bull. London Math. Soc. 32 (2000), no. 3, 311 – 315. · Zbl 1021.20012
[22] Robert M. Guralnick, Some applications of subgroup structure to probabilistic generation and covers of curves, Algebraic groups and their representations (Cambridge, 1997) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 517, Kluwer Acad. Publ., Dordrecht, 1998, pp. 301 – 320. · Zbl 0973.20011
[23] 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. · Zbl 0973.20012
[24] R. Guralnick, M. Larsen, C. Manack, Low dimensional representations of simple Lie groups. Preprint, arXiv:1009.6183. · Zbl 1247.22006
[25] R. Guralnick, G. Malle, Simple groups admit Beauville structures. Proc. Amer. Math. Soc., to appear. · Zbl 1255.20009
[26] Robert M. Guralnick and Attila Maróti, Average dimension of fixed point spaces with applications, Adv. Math. 226 (2011), no. 1, 298 – 308. · Zbl 1211.20011
[27] Robert M. Guralnick and Michael G. Neubauer, Monodromy groups of branched coverings: the generic case, Recent developments in the inverse Galois problem (Seattle, WA, 1993) Contemp. Math., vol. 186, Amer. Math. Soc., Providence, RI, 1995, pp. 325 – 352. · Zbl 0845.20002
[28] Robert Guralnick, Tim Penttila, Cheryl E. Praeger, and Jan Saxl, Linear groups with orders having certain large prime divisors, Proc. London Math. Soc. (3) 78 (1999), no. 1, 167 – 214. · Zbl 1041.20035
[29] Christoph Hering, Transitive linear groups and linear groups which contain irreducible subgroups of prime order, Geometriae Dedicata 2 (1974), 425 – 460. · Zbl 0292.20045
[30] Gerhard Hiss and Gunter Malle, Low-dimensional representations of quasi-simple groups, LMS J. Comput. Math. 4 (2001), 22 – 63. , https://doi.org/10.1112/S1461157000000796 Gerhard Hiss and Gunter Malle, Corrigenda: ”Low-dimensional representations of quasi-simple groups” [LMS J. Comput. Math. 4 (2001), 22 – 63; MR1835851 (2002b:20015)], LMS J. Comput. Math. 5 (2002), 95 – 126. · Zbl 1053.20504
[31] P. N. Hoffman and J. F. Humphreys, Projective representations of the symmetric groups, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1992. \?-functions and shifted tableaux; Oxford Science Publications. · Zbl 0777.20005
[32] I. M. Isaacs, Thomas Michael Keller, U. Meierfrankenfeld, and Alexander Moretó, Fixed point spaces, primitive character degrees and conjugacy class sizes, Proc. Amer. Math. Soc. 134 (2006), no. 11, 3123 – 3130. · Zbl 1117.20006
[33] Peter B. Kleidman, The maximal subgroups of the finite 8-dimensional orthogonal groups \?\Omega \(^{+}\)\(_{8}\)(\?) and of their automorphism groups, J. Algebra 110 (1987), no. 1, 173 – 242. · Zbl 0623.20031
[34] Peter B. Kleidman, The maximal subgroups of the Chevalley groups \?\(_{2}\)(\?) with \? odd, the Ree groups ²\?\(_{2}\)(\?), and their automorphism groups, J. Algebra 117 (1988), no. 1, 30 – 71. · Zbl 0651.20020
[35] Michael Larsen and Aner Shalev, Characters of symmetric groups: sharp bounds and applications, Invent. Math. 174 (2008), no. 3, 645 – 687. · Zbl 1166.20009
[36] M. Larsen, A. Shalev, P. Tiep, Waring problem for finite simple groups. Annals Math., to appear. · Zbl 1283.20008
[37] 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. · Zbl 0696.20004
[38] Martin W. Liebeck and Gary M. Seitz, A survey of maximal subgroups of exceptional groups of Lie type, Groups, combinatorics & geometry (Durham, 2001) World Sci. Publ., River Edge, NJ, 2003, pp. 139 – 146. · Zbl 1032.20010
[39] Frank Lübeck, Smallest degrees of representations of exceptional groups of Lie type, Comm. Algebra 29 (2001), no. 5, 2147 – 2169. · Zbl 1004.20003
[40] Frank Lübeck and Gunter Malle, (2,3)-generation of exceptional groups, J. London Math. Soc. (2) 59 (1999), no. 1, 109 – 122. · Zbl 0935.20021
[41] A. M. Macbeath, Generators of the linear fractional groups, Number Theory (Proc. Sympos. Pure Math., Vol. XII, Houston, Tex., 1967) Amer. Math. Soc., Providence, R.I., 1969, pp. 14 – 32.
[42] Gunter Malle, Die unipotenten Charaktere von ²\?\(_{4}\)(\?²), Comm. Algebra 18 (1990), no. 7, 2361 – 2381 (German). · Zbl 0721.20008
[43] Gunter Malle and B. Heinrich Matzat, Inverse Galois theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1999. · Zbl 0940.12001
[44] Gunter Malle, Jan Saxl, and Thomas Weigel, Generation of classical groups, Geom. Dedicata 49 (1994), no. 1, 85 – 116. · Zbl 0832.20029
[45] P. M. Neumann, A study of some finite permutation groups. Ph.D. thesis, Oxford, 1966.
[46] Alexander J. E. Ryba, Short proofs of embeddings into exceptional groups of Lie type, J. Algebra 249 (2002), no. 2, 402 – 418. · Zbl 1011.20017
[47] Leonard L. Scott, Matrices and cohomology, Ann. of Math. (2) 105 (1977), no. 3, 473 – 492. · Zbl 0399.20047
[48] Dan Segal and Aner Shalev, On groups with bounded conjugacy classes, Quart. J. Math. Oxford Ser. (2) 50 (1999), no. 200, 505 – 516. · Zbl 0946.20011
[49] Pham Huu Tiep and Alexander E. Zalesskii, Minimal characters of the finite classical groups, Comm. Algebra 24 (1996), no. 6, 2093 – 2167. · Zbl 0901.20031
[50] Thomas S. Weigel, Generation of exceptional groups of Lie-type, Geom. Dedicata 41 (1992), no. 1, 63 – 87. · Zbl 0758.20001
[51] Helmut Wielandt, Finite permutation groups, Translated from the German by R. Bercov, Academic Press, New York-London, 1964. · Zbl 0138.02501
[52] Alan Williamson, On primitive permutation groups containing a cycle, Math. Z. 130 (1973), 159 – 162. · Zbl 0249.20003
[53] R. Wilson, P. Walsh, J. Tripp, I. Suleiman, R. Parker, S. Norton, S. Nickerson, S. Linton, J. Bray, R. Abbott, ATLAS of Finite Group Representations - Version 3, http://brauer.maths.qmul.ac.uk/Atlas/v3/.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.