Presentations of finite simple groups: A quantitative approach

Guralnick, R.; Kantor, W.; Kassabov, M.; Lubotzky, A.

doi:10.1090/S0894-0347-08-00590-0

Presentations of finite simple groups: A quantitative approach
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by R. M. Guralnick, W. M. Kantor, M. Kassabov and A. Lubotzky;

J. Amer. Math. Soc. 21 (2008), 711-774

DOI: https://doi.org/10.1090/S0894-0347-08-00590-0

Published electronically: February 18, 2008

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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

M. Aschbacher and R. Guralnick, Some applications of the first cohomology group, J. Algebra 90 (1984), no. 2, 446–460. MR 760022, DOI 10.1016/0021-8693(84)90183-2
L. Babai, A. J. Goodman, W. M. Kantor, E. M. Luks, and P. P. Pálfy, Short presentations for finite groups, J. Algebra 194 (1997), no. 1, 79–112. MR 1461483, DOI 10.1006/jabr.1996.6980
L. Babai, W. M. Kantor, and A. Lubotsky, Small-diameter Cayley graphs for finite simple groups, European J. Combin. 10 (1989), no. 6, 507–522. MR 1022771, DOI 10.1016/S0195-6698(89)80067-8
Gilbert Baumslag, A finitely presented metabelian group with a free abelian derived group of infinite rank, Proc. Amer. Math. Soc. 35 (1972), 61–62. MR 299662, DOI 10.1090/S0002-9939-1972-0299662-4
H. Behr and J. Mennicke, A presentation of the groups $\textrm {PSL}(2,\,p)$, Canadian J. Math. 20 (1968), 1432–1438. MR 236269, DOI 10.4153/CJM-1968-144-7
Curtis D. Bennett and Sergey Shpectorov, A new proof of a theorem of Phan, J. Group Theory 7 (2004), no. 3, 287–310. MR 2062999, DOI 10.1515/jgth.2004.010
J. Bray, M. D. E. Conder, C. R. Leedham-Green and E. A. O’Brien, Short presentations for alternating and symmetric groups (preprint).
W. Burnside, Theory of groups of finite order, Dover Publications, Inc., New York, 1955. 2d ed. MR 69818
C. M. Campbell, P. P. Campbell, B. T. K. Hopson, and E. F. Robertson, On the efficiency of direct powers of $\textrm {PGL}(2,p)$, Recent advances in group theory and low-dimensional topology (Pusan, 2000) Res. Exp. Math., vol. 27, Heldermann, Lemgo, 2003, pp. 27–34. MR 2004629
Colin Campbell, George Havas, Stephen Linton, and Edmund Robertson, Symmetric presentations and orthogonal groups, The atlas of finite groups: ten years on (Birmingham, 1995) London Math. Soc. Lecture Note Ser., vol. 249, Cambridge Univ. Press, Cambridge, 1998, pp. 1–10. MR 1647409, DOI 10.1017/CBO9780511565830.006
Colin M. Campbell, George Havas, Colin Ramsay, and Edmund F. Robertson, Nice efficient presentations for all small simple groups and their covers, LMS J. Comput. Math. 7 (2004), 266–283. MR 2118175, DOI 10.1112/S1461157000001121
C. M. Campbell and E. F. Robertson, Classes of groups related to $F^{a,b,c}$, Proc. Roy. Soc. Edinburgh Sect. A 78 (1977/78), no. 3-4, 209–218. MR 577063, DOI 10.1017/S030821050000994X
C. M. Campbell and E. F. Robertson, A deficiency zero presentation for $\textrm {SL}(2,\,p)$, Bull. London Math. Soc. 12 (1980), no. 1, 17–20. MR 565476, DOI 10.1112/blms/12.1.17
C. M. Campbell, E. F. Robertson, and P. D. Williams, On presentations of $\textrm {PSL}(2,p^n)$, J. Austral. Math. Soc. Ser. A 48 (1990), no. 2, 333–346. MR 1033184
Colin M. Campbell, E. F. Robertson, and P. D. Williams, Efficient presentations for finite simple groups and related groups, Groups—Korea 1988 (Pusan, 1988) Lecture Notes in Math., vol. 1398, Springer, Berlin, 1989, pp. 65–72. MR 1032811, DOI 10.1007/BFb0086240
Robert D. Carmichael, Introduction to the theory of groups of finite order, Dover Publications, Inc., New York, 1956. MR 75938
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.
H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], Band 14, Springer-Verlag, New York-Heidelberg, 1972. MR 349820, DOI 10.1007/978-3-662-21946-1
Charles W. Curtis, Central extensions of groups of Lie type, J. Reine Angew. Math. 220 (1965), 174–185. MR 188299, DOI 10.1515/crll.1965.220.174
Dragomir Ž. Đoković, Presentations of some finite simple groups, J. Austral. Math. Soc. Ser. A 45 (1988), no. 2, 143–168. MR 951574, DOI 10.1017/S1446788700030068
Daniel Gorenstein, Richard Lyons, and Ronald Solomon, The classification of the finite simple groups. Number 3. Part I. Chapter A, Mathematical Surveys and Monographs, vol. 40, American Mathematical Society, Providence, RI, 1998. Almost simple $K$-groups. MR 1490581, DOI 10.1090/surv/040.3
R. Gramlich, C. Hoffman, W. Nickel, and S. Shpectorov, Even-dimensional orthogonal groups as amalgams of unitary groups, J. Algebra 284 (2005), no. 1, 141–173. MR 2115009, DOI 10.1016/j.jalgebra.2004.09.025
Ralf Gramlich, Corneliu Hoffman, and Sergey Shpectorov, A Phan-type theorem for $\textrm {Sp}(2n,q)$, J. Algebra 264 (2003), no. 2, 358–384. MR 1981410, DOI 10.1016/S0021-8693(03)00131-5
Robert L. Griess Jr., Schur multipliers of finite simple groups of Lie type, Trans. Amer. Math. Soc. 183 (1973), 355–421. MR 338148, DOI 10.1090/S0002-9947-1973-0338148-0
Karl W. Gruenberg, Cohomological topics in group theory, Lecture Notes in Mathematics, Vol. 143, Springer-Verlag, Berlin-New York, 1970. MR 279200, DOI 10.1007/BFb0059162
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.
R. M. Guralnick, W. M. Kantor, M. Kassabov and A. Lubotzky, Presentations of finite simple groups: a computational approach (submitted).
D. F. Holt, On the second cohomology group of a finite group, Proc. London Math. Soc. (3) 55 (1987), no. 1, 22–36. MR 887282, DOI 10.1112/plms/s3-55.1.22
Derek F. Holt, Bettina Eick, and Eamonn A. O’Brien, Handbook of computational group theory, Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2005. MR 2129747, DOI 10.1201/9781420035216
Alexander Hulpke and Ákos Seress, Short presentations for three-dimensional unitary groups, J. Algebra 245 (2001), no. 2, 719–729. MR 1863898, DOI 10.1006/jabr.2001.8943
William M. Kantor, Some topics in asymptotic group theory, Groups, combinatorics & geometry (Durham, 1990) London Math. Soc. Lecture Note Ser., vol. 165, Cambridge Univ. Press, Cambridge, 1992, pp. 403–421. MR 1200278, DOI 10.1017/CBO9780511629259.036
William M. Kantor and Ákos Seress, Black box classical groups, Mem. Amer. Math. Soc. 149 (2001), no. 708, viii+168. MR 1804385, DOI 10.1090/memo/0708
William M. Kantor and Ákos Seress, Computing with matrix groups, Groups, combinatorics & geometry (Durham, 2001) World Sci. Publ., River Edge, NJ, 2003, pp. 123–137. MR 1994963, DOI 10.1142/9789812564481_{0}007
Gregor Kemper, Frank Lübeck, and Kay Magaard, Matrix generators for the Ree groups ${}^2G_2(q)$, Comm. Algebra 29 (2001), no. 1, 407–413. MR 1842506, DOI 10.1081/AGB-100000808
Inna Korchagina and Alexander Lubotzky, On presentations and second cohomology of some finite simple groups, Publ. Math. Debrecen 69 (2006), no. 3, 341–352. MR 2273986, DOI 10.5486/pmd.2006.3590
Sava Krstić and James McCool, Presenting $\textrm {GL}_n(k\langle T\rangle )$, J. Pure Appl. Algebra 141 (1999), no. 2, 175–183. MR 1706364, DOI 10.1016/S0022-4049(98)00022-X
Charles R. Leedham-Green, The computational matrix group project, Groups and computation, III (Columbus, OH, 1999) Ohio State Univ. Math. Res. Inst. Publ., vol. 8, de Gruyter, Berlin, 2001, pp. 229–247. MR 1829483
Alexander Lubotzky, Enumerating boundedly generated finite groups, J. Algebra 238 (2001), no. 1, 194–199. MR 1822189, DOI 10.1006/jabr.2000.8650
Alexander Lubotzky, Pro-finite presentations, J. Algebra 242 (2001), no. 2, 672–690. MR 1848964, DOI 10.1006/jabr.2001.8805
Alexander Lubotzky, Finite presentations of adelic groups, the congruence kernel and cohomology of finite simple groups, Pure Appl. Math. Q. 1 (2005), no. 2, Special Issue: In memory of Armand Borel., 241–256. MR 2194724, DOI 10.4310/PAMQ.2005.v1.n2.a2
Alexander Lubotzky and Dan Segal, Subgroup growth, Progress in Mathematics, vol. 212, Birkhäuser Verlag, Basel, 2003. MR 1978431, DOI 10.1007/978-3-0348-8965-0
Wilhelm Magnus, Abraham Karrass, and Donald Solitar, Combinatorial group theory: Presentations of groups in terms of generators and relations, Interscience Publishers [John Wiley & Sons], New York-London-Sydney, 1966. MR 207802
Avinoam Mann, Enumerating finite groups and their defining relations, J. Group Theory 1 (1998), no. 1, 59–64. MR 1490158, DOI 10.1515/jgth.1998.002
G. A. Margulis, Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 17, Springer-Verlag, Berlin, 1991. MR 1090825, DOI 10.1007/978-3-642-51445-6
G. A. Miller, Abstract Definitions of all the Substitution Groups Whose Degrees do not Exceed Seven, Amer. J. Math. 33 (1911), no. 1-4, 363–372. MR 1506130, DOI 10.2307/2369998
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.
Kok Wee Phan, On groups generated by three-dimensional special unitary groups. I, J. Austral. Math. Soc. Ser. A 23 (1977), no. 1, 67–77. MR 435247, DOI 10.1017/s1446788700017353
L. Pyber, Enumerating finite groups of given order, Ann. of Math. (2) 137 (1993), no. 1, 203–220. MR 1200081, DOI 10.2307/2946623
I. Schur, Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen. J. reine angew. Math. 132 (1907) 85–137.
Ákos Seress, Permutation group algorithms, Cambridge Tracts in Mathematics, vol. 152, Cambridge University Press, Cambridge, 2003. MR 1970241, DOI 10.1017/CBO9780511546549
Jean-Pierre Serre, Le problème des groupes de congruence pour SL2, Ann. of Math. (2) 92 (1970), 489–527 (French). MR 272790, DOI 10.2307/1970630
Jean-Pierre Serre, Trees, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. Translated from the French original by John Stillwell; Corrected 2nd printing of the 1980 English translation. MR 1954121
Ken-ichi Shinoda, The conjugacy classes of the finite Ree groups of type $(F_{4})$, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), 1–15. MR 372064
Charles C. Sims, Computation with finitely presented groups, Encyclopedia of Mathematics and its Applications, vol. 48, Cambridge University Press, Cambridge, 1994. MR 1267733, DOI 10.1017/CBO9780511574702
Robert Steinberg, Lectures on Chevalley groups, Yale University, New Haven, CT, 1968. Notes prepared by John Faulkner and Robert Wilson. MR 466335
Robert Steinberg, Generators, relations and coverings of algebraic groups. II, J. Algebra 71 (1981), no. 2, 527–543. MR 630615, DOI 10.1016/0021-8693(81)90193-9
J. G. Sunday, Presentations of the groups $\textrm {SL}(2,\,m)$ and $\textrm {PSL}(2,\,m)$, Canadian J. Math. 24 (1972), 1129–1131. MR 311782, DOI 10.4153/CJM-1972-118-x
Michio Suzuki, On a class of doubly transitive groups, Ann. of Math. (2) 75 (1962), 105–145. MR 136646, DOI 10.2307/1970423
J. Tits, Les groupes de Lie exceptionnels et leur interprétation géométrique, Bull. Soc. Math. Belg. 8 (1956), 48–81 (French). MR 87889
Jacques Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, Berlin-New York, 1974. MR 470099
John S. Wilson, Finite axiomatization of finite soluble groups, J. London Math. Soc. (2) 74 (2006), no. 3, 566–582. MR 2286433, DOI 10.1112/S0024610706023106