×

zbMATH — the first resource for mathematics

The Waring problem for finite simple groups. (English) Zbl 1283.20008
Summary: The classical Waring problem deals with expressing every natural number as a sum of \(g(k)\) \(k\)-th powers. Recently there has been considerable interest in similar questions for non-Abelian groups, and simple groups in particular. Here the \(k\)-th power word can be replaced by an arbitrary group word \(w\neq 1\), and the goal is to express group elements as short products of values of \(w\).
We give a best possible and somewhat surprising solution for this Waring type problem for (non-Abelian) finite simple groups of sufficiently high order, showing that a product of length two suffices to express all elements.
Along the way we also obtain new results, possibly of independent interest, on character values in classical groups over finite fields, on regular semisimple elements lying in the image of word maps, and on products of conjugacy classes.
Our methods involve algebraic geometry and representation theory, especially Lusztig’s theory of representations of groups of Lie type.

MSC:
20D05 Finite simple groups and their classification
11P05 Waring’s problem and variants
20C33 Representations of finite groups of Lie type
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20F05 Generators, relations, and presentations of groups
20E45 Conjugacy classes for groups
20G40 Linear algebraic groups over finite fields
20E05 Free nonabelian groups
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] T. Asai, ”Unipotent characters of \({ SO}^{\pm }_{2n},\) \({ Sp}_{2n}\) and \({ SO}_{2n+1}\) over \(F_q\) with small \(q\),” Osaka J. Math., vol. 20, iss. 3, pp. 631-643, 1983. · Zbl 0516.20028
[2] A. Borel, ”On free subgroups of semisimple groups,” Enseign. Math., vol. 29, iss. 1-2, pp. 151-164, 1983. · Zbl 0533.22009
[3] A. Borel, R. Carter, C. W. Curtis, N. Iwahori, T. A. Springer, and R. Steinberg, Seminar on Algebraic Groups and Related Finite Groups, New York: Springer-Verlag, 1970, vol. 131. · Zbl 0192.36201
[4] R. W. Carter, ”Centralizers of semisimple elements in the finite classical groups,” Proc. London Math. Soc., vol. 42, iss. 1, pp. 1-41, 1981. · Zbl 0455.20035
[5] R. W. Carter, Finite Groups of Lie Type, Chichester: John Wiley & Sons Ltd., 1993. · Zbl 0900.20021
[6] R. W. Carter, ”On the representation theory of the finite groups of Lie type over an algebraically closed field of characteristic 0 [MR1170353 (93j:20034)],” in Algebra, IX, New York: Springer-Verlag, 1995, vol. 77, pp. 1-120, 235. · Zbl 0832.20020
[7] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, Atlas of Finite Groups. Maximal Subgroups and Ordinary Characters for Simple Groups, Eynsham: Oxford University Press, 1985. · Zbl 0568.20001
[8] P. Deligne, Cohomologie Étale, New York: Springer-Verlag, 1977, vol. 569. · Zbl 0345.00010
[9] P. Deligne, ”La conjecture de Weil. II,” Inst. Hautes Études Sci. Publ. Math., iss. 52, pp. 137-252, 1980. · Zbl 0456.14014
[10] P. Deligne and G. Lusztig, ”Representations of reductive groups over finite fields,” Ann. of Math., vol. 103, iss. 1, pp. 103-161, 1976. · Zbl 0336.20029
[11] E. W. Ellers and N. Gordeev, ”On the conjectures of J. Thompson and O. Ore,” Trans. Amer. Math. Soc., vol. 350, iss. 9, pp. 3657-3671, 1998. · Zbl 0910.20007
[12] H. Enomoto, ”The characters of the finite symplectic group \({ Sp}(4,\,q)\), \(q=2^f\),” Osaka J. Math., vol. 9, pp. 75-94, 1972. · Zbl 0254.20005
[13] S. Garion and A. Shalev, ”Commutator maps, measure preservation, and \(T\)-systems,” Trans. Amer. Math. Soc., vol. 361, iss. 9, pp. 4631-4651, 2009. · Zbl 1182.20015
[14] D. Gluck, ”Character value estimates for non-semisimple elements,” J. Algebra, vol. 155, iss. 1, pp. 221-237, 1993. · Zbl 0771.20009
[15] D. Gluck, ”Sharper character value estimates for groups of Lie type,” J. Algebra, vol. 174, iss. 1, pp. 229-266, 1995. · Zbl 0842.20014
[16] A. Grothendieck, ”Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I,” Inst. Hautes Études Sci. Publ. Math., iss. 20, p. 259, 1964. · Zbl 0136.15901
[17] A. Grothendieck, ”Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II,” Inst. Hautes Études Sci. Publ. Math., iss. 24, p. 231, 1965. · Zbl 0135.39701
[18] A. Grothendieck, ”Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III,” Inst. Hautes Études Sci. Publ. Math., iss. 28, p. 255, 1966. · Zbl 0144.19904
[19] R. M. Guralnick and P. H. Tiep, ”Cross characteristic representations of even characteristic symplectic groups,” Trans. Amer. Math. Soc., vol. 356, iss. 12, pp. 4969-5023, 2004. · Zbl 1062.20013
[20] G. D. James, The Representation Theory of the Symmetric Groups, New York: Springer-Verlag, 1978, vol. 682. · Zbl 0393.20009
[21] P. Kleidman and M. Liebeck, The Subgroup Structure of the Finite Classical Groups, Cambridge: Cambridge Univ. Press, 1990, vol. 129. · Zbl 0697.20004
[22] V. Landazuri and G. M. Seitz, ”On the minimal degrees of projective representations of the finite Chevalley groups,” J. Algebra, vol. 32, pp. 418-443, 1974. · Zbl 0325.20008
[23] M. Larsen and A. Shalev, ”Word maps and Waring type problems,” J. Amer. Math. Soc., vol. 22, iss. 2, pp. 437-466, 2009. · Zbl 1206.20014
[24] M. Larsen and A. Shalev, ”Characters of symmetric groups: sharp bounds and applications,” Invent. Math., vol. 174, iss. 3, pp. 645-687, 2008. · Zbl 1166.20009
[25] M. W. Liebeck, E. A. O’Brien, A. Shalev, and P. H. Tiep, ”The Ore conjecture,” J. Eur. Math. Soc. \((\)JEMS\()\), vol. 12, iss. 4, pp. 939-1008, 2010. · Zbl 1205.20011
[26] M. W. Liebeck and G. M. Seitz, Unipotent and nilpotent classes in simple algebraic groups and Lie algebras. · Zbl 1251.20001
[27] M. W. Liebeck and A. Shalev, ”Simple groups, permutation groups, and probability,” J. Amer. Math. Soc., vol. 12, iss. 2, pp. 497-520, 1999. · Zbl 0916.20003
[28] M. W. Liebeck and A. Shalev, ”Diameters of finite simple groups: sharp bounds and applications,” Ann. of Math., vol. 154, iss. 2, pp. 383-406, 2001. · Zbl 1003.20014
[29] F. Lübeck and G. Malle, ”\((2,3)\)-generation of exceptional groups,” J. London Math. Soc., vol. 59, iss. 1, pp. 109-122, 1999. · Zbl 0935.20021
[30] G. Lusztig, ”Irreducible representations of finite classical groups,” Invent. Math., vol. 43, iss. 2, pp. 125-175, 1977. · Zbl 0372.20033
[31] G. Lusztig, ”Unipotent characters of the symplectic and odd orthogonal groups over a finite field,” Invent. Math., vol. 64, iss. 2, pp. 263-296, 1981. · Zbl 0477.20023
[32] G. Lusztig, ”Unipotent characters of the even orthogonal groups over a finite field,” Trans. Amer. Math. Soc., vol. 272, iss. 2, pp. 733-751, 1982. · Zbl 0491.20034
[33] G. Lusztig, Characters of Reductive Groups over a Finite Field, Princeton, NJ: Princeton Univ. Press, 1984, vol. 107. · Zbl 0556.20033
[34] G. Malle and H. B. Matzat, Inverse Galois Theory, New York: Springer-Verlag, 1999. · Zbl 0940.12001
[35] G. Malle, J. Saxl, and T. Weigel, ”Generation of classical groups,” Geom. Dedicata, vol. 49, iss. 1, pp. 85-116, 1994. · Zbl 0832.20029
[36] C. Martinez and E. Zelmanov, ”Products of powers in finite simple groups,” Israel J. Math., vol. 96, iss. part B, pp. 469-479, 1996. · Zbl 0890.20013
[37] M. B. Nathanson, Additive Number Theory: The Classical Bases, New York: Springer-Verlag, 1996, vol. 164. · Zbl 0859.11003
[38] N. Nikolov and L. Pyber, ”Product decompositions of quasirandom groups and a Jordan type theorem,” J. Euro. Math. Soc., vol. 13, pp. 1063-1077, 2011. · Zbl 1228.20020
[39] J. Saxl and J. S. Wilson, ”A note on powers in simple groups,” Math. Proc. Cambridge Philos. Soc., vol. 122, iss. 1, pp. 91-94, 1997. · Zbl 0890.20014
[40] A. Shalev, ”Word maps, conjugacy classes, and a noncommutative Waring-type theorem,” Ann. of Math., vol. 170, iss. 3, pp. 1383-1416, 2009. · Zbl 1203.20013
[41] A. Shalev, ”Mixing and generation in simple groups,” J. Algebra, vol. 319, iss. 7, pp. 3075-3086, 2008. · Zbl 1146.20057
[42] R. Steinberg, ”A geometric approach to the representations of the full linear group over a Galois field,” Trans. Amer. Math. Soc., vol. 71, pp. 274-282, 1951. · Zbl 0045.30201
[43] R. Steinberg, ”Regular elements of semisimple algebraic groups,” Inst. Hautes Études Sci. Publ. Math., iss. 25, pp. 49-80, 1965. · Zbl 0136.30002
[44] P. H. Tiep and A. E. Zalesskii, ”Minimal characters of the finite classical groups,” Comm. Algebra, vol. 24, iss. 6, pp. 2093-2167, 1996. · Zbl 0901.20031
[45] P. H. Tiep and A. E. Zalesskii, ”Real conjugacy classes in algebraic groups and finite groups of Lie type,” J. Group Theory, vol. 8, iss. 3, pp. 291-315, 2005. · Zbl 1076.20033
[46] Y. Varshavsky, ”Lefschetz-Verdier trace formula and a generalization of a theorem of Fujiwara,” Geom. Funct. Anal., vol. 17, iss. 1, pp. 271-319, 2007. · Zbl 1131.14019
[47] J. Wilson, ”First-order group theory,” in Infinite Groups 1994, Berlin: de Gruyter, 1996, pp. 301-314. · Zbl 0866.20001
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.