Finite linear groups and bounded generation. (English) Zbl 1017.20039

Let \(p\) be a prime and \(F\) a field of characteristic \(p\). The main result of this paper is Theorem A. If \(n\) is a positive integer and \(p\) is sufficiently large, then any finite subgroup of \(\text{GL}(n,F)\) generated by its Sylow \(p\)-subgroups is a product of \(25\) Sylow \(p\)-subgroups. In the case that \(F\) is the prime field, a variation of this result was obtained by E. Hrushovski and A. Pillay [J. Reine Angew. Math. 462, 69-91 (1995; Zbl 0823.12005)]. The proof depends on a recent result of M. Larsen and R. Pink [“Finite subgroups of algebraic groups”, J. Am. Math. Soc. (to appear)] asserting that there are only finitely many simple groups other than Chevalley groups in the natural characteristic having projective representations of dimension at most \(n\). Using the classification of finite simple groups and a recent result of the reviewer [J. Algebra 220, No. 2, 531-541 (1999; Zbl 0941.20001)], \(p\) sufficiently large can be replaced by \(p\geq n+3\) (for \(n\geq 9\)) and this result is best possible. The results of Larsen-Pink (or the classification of finite simple groups) allow one to reduce to the case of Chevalley groups in characteristic \(p\). In that case, the result follows from the Bruhat decomposition.
Using the results mentioned above, the authors also confirm a conjecture of Sury and Platonov that any closed finitely generated subgroup of the direct product of \(\text{SL}(n,\mathbb{Z}_p)\) is boundedly generated (here \(\mathbb{Z}_p\) is the ring of \(p\)-adic integers).
An example is given to show that one cannot replace Sylow \(p\)-subgroup by Abelian \(p\)-subgroup in Theorem A. It is asked whether at least \(G\) (as in Theorem A for \(p\) sufficiently large) is a product of \(g(n)\) Abelian \(p\)-subgroups for some function \(g(n)\). We close this review with an example of Aschbacher and the reviewer showing that this is not true. Note that it is true for \(n=2\).
Let \(q=p^a\). Let \(G\) be the subgroup of \(\text{SL}(3,q)\) consisting of upper triangular unipotent elements such that the \((2,3)\) entry is the \(p\)-th power of the \((1,2)\) entry. This is a group of order \(q^2\). It is straightforward to check that any Abelian subgroup of \(G\) has order at most \(qp\). Thus, \(G\) is a product of \(a\) Abelian subgroups, but no fewer. Since \(a\) can be arbitrary, there is no bound on the number of Abelian subgroups needed to generate a finite subgroup (generated by its elements of order \(p\)) of \(\text{SL}(3,F)\) even if we allow \(p\) to be large. (Also submitted to MR).


20G40 Linear algebraic groups over finite fields
20E18 Limits, profinite groups
20F05 Generators, relations, and presentations of groups
20D40 Products of subgroups of abstract finite groups
Full Text: DOI


[1] M. Aschbacher, Finite Group Theory , Cambridge Stud. Adv. Math. 10 , Cambridge Univ. Press, Cambridge, 1986. · Zbl 0583.20001
[2] R. W. Carter, Conjugacy classes in the Weyl group , Compositio Math. 25 (1972), 1–59. · Zbl 0254.17005
[3] ——–, Simple Groups of Lie Type , Pure Appl. Math. 28 , Wiley, London, 1972. · Zbl 0248.20015
[4] J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, Analytic Pro-\(p\)-Groups , London Math. Soc. Lecture Note Ser. 157 , Cambridge Univ. Press, Cambridge, 1991. · Zbl 0744.20002
[5] K. Doerk and T. Hawkes, Finite Soluble Groups , de Gruyter Exp. Math. 4 , de Gruyter, Berlin, 1992. · Zbl 0753.20001
[6] R. K. Fisher, The number of non-solvable sections in linear groups , J. London Math. Soc. (2) 9 (1974/75), 80–86. · Zbl 0298.20038 · doi:10.1112/jlms/s2-9.1.80
[7] R. M. Guralnick, Small representations are completely reducible , J. Algebra 220 (1999), 531–541. · Zbl 0941.20001 · doi:10.1006/jabr.1999.7963
[8] E. Hrushovski and A. Pillay, Definable subgroups of algebraic groups over finite fields , J. Reine Angew. Math. 462 (1995), 69–91. · Zbl 0823.12005
[9] I. M. Isaacs, Character Theory of Finite Groups , Pure Appl. Math. 69 , Academic Press, New York, 1976. · Zbl 0337.20005
[10] P. B. Kleidman and M. W. Liebeck, The Subgroup Structure of the Finite Classical Groups , London Math. Soc. Lecture Note Ser. 129 , Cambridge Univ. Press, Cambridge, 1990. · Zbl 0697.20004
[11] M. J. Larsen and R. Pink, Finite subgroups of algebraic groups , to appear in J. Amer. Math. Soc., http://www.math.ethz.ch/ pink/preprints.html.
[12] R. Lawther and M. W. Liebeck, On the diameter of a Cayley graph of a simple group of Lie type based on a conjugacy class , J. Combin. Theory Ser. A 83 (1998), 118–137. · Zbl 0911.05035 · doi:10.1006/jcta.1998.2869
[13] A. Lubotzky, Subgroup growth and congruence subgroups , Invent. Math. 119 (1995), 267–295. · Zbl 0848.20036 · doi:10.1007/BF01245183
[14] M. V. Nori, On subgroups of \(\GL_n(\mathbb F_p)\) , Invent. Math. 88 (1987), 257–275. · Zbl 0632.20030 · doi:10.1007/BF01388909
[15] A. Yu. Olshanskii, The number of generators and orders of abelian subgroups of finite \(p\)-groups , Math. Notes 23 (1978), 183–185. · Zbl 0403.20014 · doi:10.1007/BF01651428
[16] V. P. Platonov and A. S. Rapinchuk, Abstract properties of \(S\)-arithmetic groups and the congruence subgroup problem , Russian Akad. Sci. Izv. Math. 40 (1993), 455–476. · Zbl 0785.20025 · doi:10.1070/IM1993v040n03ABEH002173
[17] V. P. Platonov and B. Sury, Adelic profinite groups , J. Algebra 193 (1997), 757–763. · Zbl 0884.20018 · doi:10.1006/jabr.1996.7011
[18] L. Pyber, Bounded generation and subgroup growth , preprint, 1999. · Zbl 1041.20016 · doi:10.1112/S0024609301008657
[19] M. Suzuki, Group Theory, I , Grundlehren Math. Wiss. 247 , Springer, Berlin, 1982.
[20] B. Weisfeiler, Post-classification version of Jordan’s theorem on finite linear groups , Proc. Nat. Acad. Sci. U.S.A. 81 (1984), 5278–5279. JSTOR: · Zbl 0542.20026 · doi:10.1073/pnas.81.16.5278
[21] ——–, On the size and structure of finite linear groups ,
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.