Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)



Presentations of finite simple groups: A quantitative approach

Authors: R. M. Guralnick, W. M. Kantor, M. Kassabov and A. Lubotzky
Journal: J. Amer. Math. Soc. 21 (2008), 711-774
MSC (2000): Primary 20D06, 20F05; Secondary 20J06
Published electronically: February 18, 2008
MathSciNet review: 2393425
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: There is a constant $ C_0$ such that all nonabelian finite simple groups of rank $ n$ over $ \mathbb{F}_q$, with the possible exception of the Ree groups $ ^2G_2(3^{2e+1})$, have presentations with at most $ C_0$ generators and relations and total length at most $ C_0(\log n +\log q)$. As a corollary, we deduce a conjecture of Holt: there is a constant $ C$ such that $ \dim H^2(G,M) \leq C\dim M$ for every finite simple group $ G$, every prime $ p$ and every irreducible $ {\mathbb{F}}_p G $-module $ M$.

References [Enhancements On Off] (What's this?)

  • [AG] M. Aschbacher and R. Guralnick, Some applications of the first cohomology group. J. Algebra 90 (1984) 446-460. MR 760022 (86m:20060)
  • [BGKLP] L. Babai, A. J. Goodman, W. M. Kantor, E. M. Luks and P. P. Pálfy, Short presentations for finite groups. J. Algebra 194 (1997) 79-112. MR 1461483 (98h:20044)
  • [BKL] L. Babai, W. M. Kantor and A. Lubotzky, Small diameter Cayley graphs for finite simple groups. Eur. J. Comb. 10 (1989) 507-522. MR 1022771 (91a:20038)
  • [Bau] G. Baumslag, A finitely presented metabelian group with a free abelian derived group of infinite rank. Proc. Amer. Math. Soc. 35 (1972) 61-62. MR 0299662 (45:8710)
  • [BM] H. Behr and J. Mennicke, A presentation of the groups $ {\operatorname{PSL}}(2,p)$. Canad. J. Math. 20 (1968) 1432-1438. MR 0236269 (38:4566)
  • [BS] C. D. Bennett and S. Shpectorov, A new proof of Phan's theorem. J. Group Theory 7 (2004) 287-310. MR 2062999 (2005k:57004)
  • [BCLO] J. Bray, M. D. E. Conder, C. R. Leedham-Green and E. A. O'Brien, Short presentations for alternating and symmetric groups (preprint).
  • [Bur] W. Burnside, Theory of Groups of Finite Order, 2nd ed., Cambridge Univ. Press, Cambridge, 1911. MR 0069818 (16:1086c)
  • [CCHR] C. M. Campbell, P. P. Campbell, B. T. K. Hopson and E. F. Robertson, On the efficiency of direct powers of $ {\rm PGL}(2,p)$, pp. 27-34 in: Recent advances in group theory and low-dimensional topology (Pusan, 2000; Eds. J. Rae Cho and J. Mennicke). Heldermann, Lemgo, 2003. MR 2004629 (2004g:20044)
  • [CHLR] C. M.  Campbell, G. Havas, S. Linton and E. F. Robertson, Symmetric presentations and orthogonal groups, pp. 1-10 in: The atlas of finite groups: ten years on (Birmingham, 1995; Eds. R. Curtis and R. Wilson), Lond. Math. Soc. Lecture Note 249. Cambridge Univ. Press, Cambridge, 1998. MR 1647409 (99m:20112)
  • [CHRR] C. M.  Campbell, G. Havas, C. Ramsay and E. F. Robertson, Nice efficient presentations for all small simple groups and their covers. Lond. Math. Soc. J. Comput. Math. 7 (2004) 266-283. MR 2118175 (2006g:20046)
  • [CR1] C. M. Campbell and E. F. Robertson, Classes of groups related to $ F^{a,b,c}$. Proc. Roy. Soc. Edinburgh A78 (1977/78) 209-218. MR 0577063 (58:28181)
  • [CR2] C. M. Campbell and E. F. Robertson, A deficiency zero presentation for $ {\rm SL}(2,\,p)$. Bull. Lond. Math. Soc. 12 (1980) 17-20. MR 565476 (81d:20023)
  • [CRW1] C. M. Campbell, E. F. Robertson and P. D. Williams, On presentations of $ \operatorname{PSL}(2,p^n)$. J. Austral. Math. Soc. 48 (1990) 333-346. MR 1033184 (90j:20066)
  • [CRW2] C. M. Campbell, E. F. Robertson and P. D. Williams, Efficient presentations for finite simple groups and related groups, pp. 65-72 in: Groups-Korea, 1988 (Pusan, 1988; Eds. Y. Gheel Baik et al.). Springer, Berlin, 1989. MR 1032811 (91a:20031)
  • [Carm] R. D. Carmichael, Introduction to the theory of groups of finite order. Ginn, Boston, 1937. MR 0075938 (17:823a)
  • [Cox] H. S. M. Coxeter, Abstract groups of the form $ V^k_1 = V^3_j = (V_iV_j)^2 = 1$. J. Lond. Math. Soc. 9 (1934) 213-219.
  • [CoMo] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete B. 14. Springer, New York-Heidelberg, 1972. MR 0349820 (50:2313)
  • [Cur] C. W. Curtis, Central extensions of groups of Lie type. J. reine angew. Math. 220 (1965) 174-185. MR 0188299 (32:5738)
  • [Do] D. Z. Djokovic, Presentations of some finite simple groups. J. Austral. Math. Soc. 45 (1988) 143-168. MR 951574 (89k:20023)
  • [GLS] D. Gorenstein, R. Lyons and R. Solomon, The classification of the finite simple groups. Number 3. Part I. Chapter A. Almost simple K-groups. Amer. Math. Soc., Providence, 1998. MR 1490581 (98j:20011)
  • [GHNS] R. Gramlich, C. Hoffman, W. Nickel and S. Shpectorov, Even-dimensional orthogonal groups as amalgams of unitary groups. J. Algebra 284 (2005) 141-173. MR 2115009 (2006f:20058)
  • [GHS] R. Gramlich, C. Hoffman and S. Shpectorov, A Phan-type theorem for $ \operatorname{Sp}(2n, q)$. J. Algebra 264 (2003) 358-384. MR 1981410 (2004i:20088)
  • [Gr] R. L. Griess, Jr., Schur multipliers of finite simple groups of Lie type. Trans. Amer. Math. Soc. 183 (1973) 355-421. MR 0338148 (49:2914)
  • [Gru] K. W. Gruenberg, Cohomological topics in group theory. Lecture Notes in Mathematics, 143. Springer, Berlin-New York, 1970. MR 0279200 (43:4923)
  • [GKKL1] R. M. Guralnick, W. M. Kantor, M. Kassabov and A. Lubotzky, Presentations of finite simple groups: a cohomological and profinite approach. Groups, Geometry and Dynamics 1 (2007) 469-523.
  • [GKKL2] R. M. Guralnick, W. M. Kantor, M. Kassabov and A. Lubotzky, Presentations of finite simple groups: a computational approach (submitted).
  • [Ho] D. F. Holt, On the second cohomology group of a finite group. Proc. Lond. Math. Soc. (3) 55 (1987) 22-36. MR 887282 (88c:20063)
  • [HEO] D. F. Holt, B.  Eick and E. A. O'Brien, Handbook of computational group theory. Chapman & Hall, Boca Raton, 2005. MR 2129747 (2006f:20001)
  • [HS] A. Hulpke and Á. Seress, Short presentations for three-dimensional unitary groups. J. Algebra 245 (2001) 719-729. MR 1863898 (2002m:20079)
  • [Ka] W. M. Kantor, Some topics in asymptotic group theory, pp. 403-421 in: Groups, Combinatorics and Geometry (Durham, 1990; Eds. M. W. Liebeck and J. Saxl), Lond. Math. Soc. Lecture Note 165. Cambridge Univ. Press, Cambridge, 1992. MR 1200278 (94a:20031)
  • [KS1] W. M. Kantor and Á. Seress, Black box classical groups. Memoirs Amer. Math. Soc. 708 (2001). MR 1804385 (2001m:68066)
  • [KS2] W. M. Kantor and Á. Seress, Computing with matrix groups, pp. 123-137 in: Groups, combinatorics and geometry (Durham, 2001; Eds. A. A. Ivanov et al.). World Sci. Publ., River Edge, NJ, 2003. MR 1994963 (2004k:20098)
  • [KLM] G. Kemper, F. Lübeck and K. Magaard, Matrix generators for the Ree groups $ ^2 G_2(q)$. Comm. Alg. 29 (2001) 407-413. MR 1842506 (2002e:20025)
  • [KoLu] I. Korchagina and A. Lubotzky, On presentations and second cohomology of some finite simple groups. Publ. Math. Debrecen 69 (2006) 341-352. MR 2273986 (2007i:20057)
  • [KM] S. Krstic and J. McCool, Presenting $ \operatorname{GL}_n(k \langle T \rangle)$. J. Pure Appl. Algebra 141 (1999) 175-183. MR 1706364 (2000f:19001)
  • [LG] C. R. Leedham-Green, The computational matrix group project, pp. 229-247 in: Groups and Computation III (Eds. W. M. Kantor and Á. Seress). deGruyter, Berlin-New York, 2001. MR 1829483 (2002d:20084)
  • [Lub1] A. Lubotzky, Enumerating boundedly generated finite groups. J. Algebra 238 (2001) 194-199. MR 1822189 (2002d:20036)
  • [Lub2] A. Lubotzky, Pro-finite presentations. J. Algebra 242 (2001) 672-690. MR 1848964 (2002i:20045)
  • [Lub3] A. Lubotzky, Finite presentations of adelic groups, the congruence kernel and cohomology of finite simple groups. Pure Appl. Math. Q. 1 (2005) 241-256. MR 2194724 (2007k:20106)
  • [LS] A. Lubotzky and D. Segal, Subgroup growth. Birkhäuser, Basel, 2003. MR 1978431 (2004k:20055)
  • [MKS] W. Magnus, A. Karrass and D. Solitar, Combinatorial group theory; presentations of groups in terms of generators and relations. Interscience, New York, 1966. MR 0207802 (34:7617)
  • [Man] A. Mann, Enumerating finite groups and their defining relations. J. Group Theory 1 (1998) 59-64. MR 1490158 (98m:20024)
  • [Mar] G. A. Margulis, Discrete subgroups of semisimple Lie groups. Springer, Berlin, 1991. MR 1090825 (92h:22021)
  • [Mil] G. A. Miller, Abstract definitions of all the substitution groups whose degrees do not exceed seven. Amer. J. Math. 33 (1911) 363-372. MR 1506130
  • [Mo] E. H. Moore, Concerning the abstract groups of order $ k!$ and $ \frac{1}{2}k!$ holohedrically isomorphic with the symmetric and the alternating substitution groups on $ k$ letters. Proc. Lond. Math. Soc. 28 (1897) 357-366.
  • [Ph] K. W. Phan, On groups generated by three-dimensional special unitary groups I. J. Austral. Math. Soc. Ser. A23 (1977) 67-77. MR 0435247 (55:8207)
  • [Py] L. Pyber, Enumerating finite groups of given order. Ann. of Math. 137 (1993) 203-220. MR 1200081 (93m:11097)
  • [Sch] I. Schur, Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen. J. reine angew. Math. 132 (1907) 85-137.
  • [Ser] Á. Seress, Permutation group algorithms. Cambridge Univ. Press, Cambridge, 2003. MR 1970241 (2004c:20008)
  • [Ser1] J.-P. Serre, Le problème des groupes de congruence pour SL2. Ann. of Math. 92 (1970) 489-527. MR 0272790 (42:7671)
  • [Ser2] J.-P. Serre, Trees. Springer, Berlin, 2003. MR 1954121 (2003m:20032)
  • [Shi] K. Shinoda, The conjugacy classes of the finite Ree groups of type $ F_4$. J. Fac. Sci. Univ. Tokyo 22 (1975) 1-15. MR 0372064 (51:8281)
  • [Sim] C. C. Sims, Computation with finitely presented groups. Cambridge Univ. Press, Cambridge, 1994. MR 1267733 (95f:20053)
  • [St1] R. Steinberg, Lectures on Chevalley groups (mimeographed notes). Yale Univ., 1967. MR 0466335 (57:6215)
  • [St2] R. Steinberg. Generators, relations and coverings of algebraic groups, II. J. Algebra 71 (1981) 527-543. MR 630615 (83d:14025)
  • [Sun] J. G. Sunday, Presentations of the groups $ {\rm SL}(2,\,m)$ and $ {\rm PSL}(2,\,m)$. Canad. J. Math. 24 (1972) 1129-1131. MR 0311782 (47:344)
  • [Suz] M. Suzuki, On a class of doubly transitive groups. Ann. of Math. 75 (1962) 105-145. MR 0136646 (25:112)
  • [Ti1] J. Tits, Les groupes de Lie exceptionnels et leur interprétation géométrique. Bull. Soc. Math. Belg. 8 (1956) 48-81. MR 0087889 (19:430d)
  • [Ti2] J. Tits, Buildings of spherical type and finite BN-pairs. Springer, Berlin-New York, 1974. MR 0470099 (57:9866)
  • [Wi] J. S. Wilson, Finite axiomatization of finite soluble groups. JLMS 74 (2006) 566-582. MR 2286433 (2007i:20036)

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 20D06, 20F05, 20J06

Retrieve articles in all journals with MSC (2000): 20D06, 20F05, 20J06

Additional Information

R. M. Guralnick
Affiliation: Department of Mathematics, University of Southern California, Los Angeles, California 90089-2532

W. M. Kantor
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403

M. Kassabov
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853-4201

A. Lubotzky
Affiliation: Department of Mathematics, Hebrew University, Givat Ram, Jerusalem 91904, Israel

Received by editor(s): February 22, 2006
Published electronically: February 18, 2008
Additional Notes: The authors were partially supported by NSF grants DMS 0140578, DMS 0242983, DMS 0600244 and DMS 0354731. The authors are grateful for the support and hospitality of the Institute for Advanced Study, where this research was carried out. The research by the last author was also supported by the Ambrose Monell Foundation and the Ellentuck Fund.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society