The Margulis-Platonov conjecture for \(\text{SL}_{1,D}\) and 2-generation of finite simple groups. (English) Zbl 1101.20027
Let \(K\) be a global field, \(G\) a simple simply connected algebraic group defined over \(K\), and \(A\) the set of all nonarchimedean valuations \(v\) of \(K\), such that \(G\) is anisotropic over the completion \(K_v\) of \(K\) with respect to the topology of \(v\). It is known that \(A\) is finite [see e.g. Sect. 9.1 of V. P. Platonov and A. S. Rapinchuk’s book Algebraic groups and number theory. Moscow: Nauka (1991; Zbl 0732.20027)]. Denote by \(G(L)\) the group of rational points of \(G\) over an arbitrary extension \(L\) of \(K\), and by \(G_A\) the topological group product \(\prod G(K_v)\), indexed by \(A\). Also, let \(\delta\colon G(K)\to G_A\) be the diagonal map.
The Margulis-Platonov conjecture states that for any noncentral normal subgroup \(H\) of \(G(K)\) there exists an open normal subgroup \(W\) of \(G_A\) whose pre-image in \(G(K)\) coincides with \(H\); in particular, if \(A\) is empty, then \(H\) does not possess proper noncentral normal subgroups. This conjecture has been proved in a number of cases including the one of \(G=\text{SL}_1(D)\), where \(D\) is a finite-dimensional central division \(K\)-algebra. The proofs of the obtained results rely heavily on the classification of finite simple groups [see Y. Segev and G. M. Seitz, Pac. J. Math. 202, No. 1, 125-225 (2002; Zbl 1058.20015)].
The paper under review gives a new (considerably shorter) proof of the Margulis-Platonov conjecture, for \(G=\text{SL}_1(D)\), in which the use of this classification is limited to its consequence that finite simple groups are generated by two elements.
The Margulis-Platonov conjecture states that for any noncentral normal subgroup \(H\) of \(G(K)\) there exists an open normal subgroup \(W\) of \(G_A\) whose pre-image in \(G(K)\) coincides with \(H\); in particular, if \(A\) is empty, then \(H\) does not possess proper noncentral normal subgroups. This conjecture has been proved in a number of cases including the one of \(G=\text{SL}_1(D)\), where \(D\) is a finite-dimensional central division \(K\)-algebra. The proofs of the obtained results rely heavily on the classification of finite simple groups [see Y. Segev and G. M. Seitz, Pac. J. Math. 202, No. 1, 125-225 (2002; Zbl 1058.20015)].
The paper under review gives a new (considerably shorter) proof of the Margulis-Platonov conjecture, for \(G=\text{SL}_1(D)\), in which the use of this classification is limited to its consequence that finite simple groups are generated by two elements.
Reviewer: Ivan D. Chipchakov (Sofia)
MSC:
| 20G30 | Linear algebraic groups over global fields and their integers |
| 16K20 | Finite-dimensional division rings |
| 20D05 | Finite simple groups and their classification |
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 11E57 | Classical groups |
| 16U60 | Units, groups of units (associative rings and algebras) |
Keywords:
finite dimensional central division algebras; global fields; simple simply connected algebraic groups; anisotropic groups; commuting graphs of finite groups; finite simple groupsReferences:
| [1] | Cassels, J.W.S., Frölich, A. (ed.): Algebraic number theory. Proc. of an Instructional Conf. Organized by the London Math. Soc., Academic Press, London and New York, 1967 · Zbl 0153.07403 |
| [2] | Aschbacher, M., Guralnick, R.: Some applications of the first cohomology group. J. Algebra 90, 446–460 (1984) · Zbl 0554.20017 · doi:10.1016/0021-8693(84)90183-2 |
| [3] | Bergelson, V., Shapiro, D.B.: Multiplicative subgroups of finite index in a ring. Proc. AMS 116, 885–896 (1992) · Zbl 0784.12002 · doi:10.1090/S0002-9939-1992-1095220-5 |
| [4] | Bourbaki, N.: Algèbre commutative. ch. V-VI, Masson, Paris, 1985 · Zbl 0547.13002 |
| [5] | Faith, C.: On conjugates in division rings. Canad. J. Math. 10, 374–380 (1958) · Zbl 0082.03203 · doi:10.4153/CJM-1958-036-7 |
| [6] | Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I. Publ. math. IHES 20, 259 (1964) |
| [7] | Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II. Publ. math. IHES 24, 231 (1965) |
| [8] | Koch, H.: Number Theory: Algebraic Numbers and Functions. Graduate Studies in Mathematics, vol. 24, AMS, 2000 · Zbl 0953.11001 |
| [9] | Liebeck, M.W., Shalev, A.: The probability of generating a finite simple group. Geom. Dedicata 56, 103–113 (1995) · Zbl 0836.20068 · doi:10.1007/BF01263616 |
| [10] | Lubotzky, A., Segal, D.: Subgroup Growth. Birkhäuser, 2003 · Zbl 1071.20033 |
| [11] | Malle, G., Saxl, J., Weigel, T.: Generation of classical groups. Geom. Dedicata 49, 85–116 (1994) · Zbl 0832.20029 · doi:10.1007/BF01263536 |
| [12] | Margulis, G.A.: Finiteness of quotients of discrete subgroups. Funct. Anal. Appl. 13, 178–187 (1979) · Zbl 0434.22013 |
| [13] | Platonov, V.P., Rapinchuk, A.S.: Algebraic Groups and Number Theory. Academic Press, 1993 · Zbl 0732.20027 |
| [14] | Prasad, G.: Strong approximation for semi-simple groups over function fields. Ann. Math. 105, 553–572 (1977) · Zbl 0348.22006 · doi:10.2307/1970924 |
| [15] | Raghunathan, M.S.: On the group of norm 1 elements in a division algebra. Math. Ann. 279, 457–484 (1988) · Zbl 0639.20026 · doi:10.1007/BF01456282 |
| [16] | Rapinchuk, A., Potapchik, A.: Normal subgroups of SL1 , D and the classification of finite simple groups. Proc. Indian Acad. Sci. (Math. Sci.) 106, 329–368 (1996) · Zbl 0879.20027 · doi:10.1007/BF02837693 |
| [17] | Rapinchuk, A.S., Segev, Y.: Valuation-like maps and the congruence subgroup property. Invent. Math. 144, 571–607 (2001) · Zbl 0999.16016 · doi:10.1007/s002220100136 |
| [18] | Rapinchuk, A.S., Segev, Y., Seitz, G.M.: Finite quotients of the multiplicative group of a finite dimensional division algebra are solvable. J. AMS 15, 929–978 (2002) · Zbl 1008.16018 |
| [19] | Riehm, C.: The norm 1 group of a p-adic division algebra. Am. J. Math. 92, 499–523 (1970) · Zbl 0199.37601 · doi:10.2307/2373336 |
| [20] | Segev, Y.: On finite homomorphic images of the multiplicative group of a division algebra. Ann. Math. 149, 219–251 (1999) · Zbl 0935.20009 · doi:10.2307/121024 |
| [21] | Segev, Y.: Some applications of Wedderburn’s factorization theorem. Bull. Austral. Math. Soc. 59, 105–110 (1999) · Zbl 0927.16011 · doi:10.1017/S0004972700032640 |
| [22] | Segev, Y.: The commuting graph of minimal nonsolvable groups. Geom. Ded. 88, 55–66 (2001) · Zbl 0994.20012 · doi:10.1023/A:1013180005982 |
| [23] | Segev, Y., Seitz, G.M.: Anisotropic groups of type An and the commuting graph of finite simple groups. Pacific J. Math. 202, 125–226 (2002) · Zbl 1058.20015 · doi:10.2140/pjm.2002.202.125 |
| [24] | Turnwald, G.: Multiplicative subgroups of finite index in rings. Proc. AMS 120, 377–381 (1994) · Zbl 0795.12004 · doi:10.1090/S0002-9939-1994-1215206-9 |
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.
