Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics
Tatiana Bandman; Shelly Garion; Boris Kunyavskiĭ
Open Mathematics (2014)
- Volume: 12, Issue: 2, page 175-211
- ISSN: 2391-5455
Access Full Article
topAbstract
topHow to cite
topTatiana Bandman, Shelly Garion, and Boris Kunyavskiĭ. "Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics." Open Mathematics 12.2 (2014): 175-211. <http://eudml.org/doc/269029>.
@article{TatianaBandman2014,
abstract = {We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods: group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.},
author = {Tatiana Bandman, Shelly Garion, Boris Kunyavskiĭ},
journal = {Open Mathematics},
keywords = {Matrix groups; Finite simple groups; Special linear group; Word map; Trace map; Arithmetic dynamics; Periodic points; Finite fields; Lang-Weil estimate; matrix groups; matrix equations; finite simple groups; Ore conjecture; special linear groups; word maps; trace maps; arithmetic dynamics; periodic points; finite fields; Lang-Weil estimates},
language = {eng},
number = {2},
pages = {175-211},
title = {Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics},
url = {http://eudml.org/doc/269029},
volume = {12},
year = {2014},
}
TY - JOUR
AU - Tatiana Bandman
AU - Shelly Garion
AU - Boris Kunyavskiĭ
TI - Equations in simple matrix groups: algebra, geometry, arithmetic, dynamics
JO - Open Mathematics
PY - 2014
VL - 12
IS - 2
SP - 175
EP - 211
AB - We present a survey of results on word equations in simple groups, as well as their analogues and generalizations, which were obtained over the past decade using various methods: group-theoretic and coming from algebraic and arithmetic geometry, number theory, dynamical systems and computer algebra. Our focus is on interrelations of these machineries which led to numerous spectacular achievements, including solutions of several long-standing problems.
LA - eng
KW - Matrix groups; Finite simple groups; Special linear group; Word map; Trace map; Arithmetic dynamics; Periodic points; Finite fields; Lang-Weil estimate; matrix groups; matrix equations; finite simple groups; Ore conjecture; special linear groups; word maps; trace maps; arithmetic dynamics; periodic points; finite fields; Lang-Weil estimates
UR - http://eudml.org/doc/269029
ER -
References
top- [1] Abért M., On the probability of satisfying a word in a group, J. Group Theory, 2006, 5), 685–694 Zbl1130.20052
- [2] Adolphson A., Sperber S., On the degree of the L-functions associated with an exponential sum, Compositio Math., 1988, 68(2), 125–159 Zbl0665.12022
- [3] Amitsur S.A., The T-ideals of the free ring, J. London Math. Soc., 1955, 30(4), 470–475 Zbl0064.26506
- [4] Arzhantsev I.V., Petravchuk A.P., Closed polynomials and saturated subalgebras of polynomial algebras, Ukrainian Math. J., 2007, 59(12), 1783–1790 Zbl1164.13302
- [5] Arzhantseva G.N., Ol’shanskii A.Yu., Generality of the class of groups in which subgroups with a lesser number of generators are free, Math. Notes, 1996, 59(3–4), 350–355
- [6] Baer R., Engelsche Elemente Noetherscher Gruppen, Math. Ann., 1957, 133(3), 256–270
- [7] Bandman T., Borovoi M., Grunewald F., Kunyavskiĭ B., Plotkin E., Engel-like characterization of radicals in finite dimensional Lie algebras and finite groups, Manuscripta Math., 2006, 119(4), 465–481 Zbl1174.17017
- [8] Bandman T., Garion S., Surjectivity and equidistribution of the word xayb on PSL (2; q) and SL (2; q), Internat. J. Algebra Comput., 2012, 22(2), #1250017 Zbl1255.20010
- [9] Bandman T., Garion S., Grunewald F., On the surjectivity of Engel words on PSL (2; q), Groups Geom. Dyn., 2012, 6(3), 409–439 Zbl1261.14010
- [10] Bandman T., Gordeev N., Kunyavskiĭ B., Plotkin E., Equations in simple Lie algebras, J. Algebra, 2012, 355, 67–79 Zbl1297.17003
- [11] Bandman T., Greuel G.-M., Grunewald F., Kunyavskiĭ B., Pfister G., Plotkin E., Two-variable identities for finite solvable groups, C. R. Acad. Sci. Paris, 2003, 337(9), 581–586 Zbl1047.20014
- [12] Bandman T., Greuel G.-M., Grunewald F., Kunyavskiĭ B., Pfister G., Plotkin E., Identities for finite solvable groups and equations in finite simple groups, Compos. Math., 2006, 142(3), 734–764 Zbl1112.20016
- [13] Bandman T., Grunewald F., Kunyavskiĭ B., Geometry and arithmetic of verbal dynamical systems on simple groups, Groups Geom. Dyn., 2010, 4(4), 607–655 Zbl1276.14037
- [14] Bandman T., Kunyavskiĭ B., Criteria for equidistribution of solutions of word equations in SL (2), J. Algebra, 2013, 382, 282–302 Zbl1292.20049
- [15] Blanc J., Groupes de Cremona, connexité et simplicité, Ann. Sci. Éc. Norm. Supér., 2010, 43(2), 357–364
- [16] Bodin A., Dèbes P., Najib S., Indecomposable polynomials and their spectrum, Acta Arith., 2009, 139(1), 79–100 Zbl1228.12002
- [17] Borel A., On free subgroups of semisimple groups, Enseign. Math., 1983, 29(1–2), 151–164 Zbl0533.22009
- [18] Borisov A., Sapir M., Polynomial maps over finite fields and residual finiteness of mapping tori of group endomorphisms, Invent. Math., 2005, 160(2), 341–356 Zbl1083.14023
- [19] Borisov A., Sapir M., Polynomial maps over p-adics and redisual properties of mapping tori of group endomorphisms, Int. Math. Res. Not. IMRN, 2009, 16, 3002–3015 Zbl1183.20031
- [20] Bray J.N., Wilson J.S., Wilson R.A., A characterization of finite soluble groups by laws in two variables, Bull. London Math. Soc., 2005, 37(2), 179–186 Zbl1075.20008
- [21] Breuillard E., Green B., Guralnick R., Tao T., Strongly dense free subgroups of semisimple algebraic groups, Israel J. Math., 2012, 192(1), 347–379 Zbl1266.20060
- [22] Cantat S., Lamy S., Normal subgroups in the Cremona group, Acta Math., 2013, 210(1), 31–94 Zbl1278.14017
- [23] Cargo D.P., de Launey W., Liebeck M.W., Stafford R.M., Short two-variable identities for finite groups, J. Group Theory, 2008, 11(5), 675–690 Zbl1162.20021
- [24] Casals-Ruiz M., Kazachkov I., On Systems of Equations over Free Partially Commutative Groups, Mem. Amer. Math. Soc., 212(999), American Mathematical Society, Providence, 2011
- [25] Connes A., Schwarz A., Matrix Vieta theorem revisited, Lett. Math. Phys., 1997, 39(4), 349–353 Zbl0874.15010
- [26] Deligne P., Sullivan D., Division algebras and the Hausdorff-Banach-Tarski paradox, Enseign. Math., 1983, 29(1–2), 145–150 Zbl0521.57035
- [27] Digne F., Michel J., Representations of Finite Groups of Lie Type, London Math. Soc. Stud. Texts, 21, Cambridge University Press, Cambridge, 1991 Zbl0815.20014
- [28] Dixon J.D., The probability of generating the symmetric group, Math. Z., 1969, 110(3), 199–205 Zbl0176.29901
- [29] Droste M., Truss J.K., On representing words in the automorphism group of the random graph, J. Group Theory, 2006, 9(6), 815–836 Zbl1122.20015
- [30] Elkasapy A., Thom A., About Goto’s method showing surjectivity of word maps, preprint available at http://arxiv.org/abs/1207.5596 Zbl1320.20033
- [31] Ellers E.W., Gordeev N., Gauss decomposition with prescribed semisimple part in classical Chevalley groups, Comm. Algebra, 1994, 22(14), 5935–5950 Zbl0821.20028
- [32] Ellers E.W., Gordeev N., Gauss decomposition with prescribed semisimple part in Chevalley groups II, Exceptional cases, Comm. Algebra, 1995, 23(8), 3085–3098 Zbl0838.20054
- [33] Ellers E.W., Gordeev N., Gauss decomposition with prescribed semisimple part in Chevalley groups III, Finite twisted groups, Comm. Algebra, 1996, 24(14), 4447–4475 Zbl0887.20022
- [34] Ellers E.W., Gordeev N., On the conjectures of J. Thompson and O. Ore, Trans. Amer. Math. Soc., 1998, 350(9), 3657–3671 Zbl0910.20007
- [35] Etingof P., Gelfand I., Retakh V., Factorization of differential operators, quasideterminants, and nonabelian Toda field equations, Math. Res. Lett., 1997, 4(2–3), 413–425 Zbl0959.37054
- [36] Formanek E., Central polynomials for matrix rings, J. Algebra, 1972, 23(1), 129–132 Zbl0242.15004
- [37] Fricke R., Über die Theorie der automorphen Modulgruppen, Nachr. Akad. Wiss. Göttingen, 1896, 91–101
- [38] Fricke R., Klein F., Vorlesungen über die Theorie der Automorphen Funktionen, 1 and 2, Teubner, Leipzig, 1897 and 1912
- [39] Fuchs D., Schwarz A., Matrix Vieta theorem, In: Lie groups and Lie algebras: E.B. Dynkin’s Seminar, Amer. Math. Soc. Transl. Ser. 2, 169, American Mathematical Society, Providence, 1995, 15–22
- [40] Fujiwara K., Rigid geometry, Lefschetz-Verdier trace formula and Deligne’s conjecture, Invent. Math., 1997, 127(3), 489–533 Zbl0920.14005
- [41] Garion S., Shalev A., Commutator maps, measure preservation, and T-systems, Trans. Amer. Math. Soc., 2009, 361(9), 4631–4651 Zbl1182.20015
- [42] Gelfand I., Retakh V., Noncommutative Vieta theorem and symmetric functions, In: The Gelfand Mathematical Seminars, 1993–1995, Gelfand Math. Sem., Birkhäuser, Boston, 1996, 93–100 Zbl0865.05074
- [43] Gelfand I., Retakh V., Quasideterminants I, Selecta Math. (N.S.), 1997, 3(4), 517–546
- [44] Gelfand S., On the number of solutions of a quadratic equation, In: Globus: General Mathematical Seminar, 1, Independent University of Moscow, Moscow, 2004, 124–133 (in Russian)
- [45] Ghorpade S.R., Lachaud G., Number of solutions of equations over finite fields and a conjecture of Lang and Weil, In: Number Theory and Discrete Mathematics, Chandigarh, October 2–6, 2000, Trends Math., Birkhäuser, Basel, 2002, 269–291 Zbl1080.11049
- [46] Ghorpade S.R., Lachaud G., Étale cohomology, Lefschetz theorems and number of points of singular varieties over finite fields, Mosc. Math. J., 2002, 2(3), 589–631; 2009, 9 (2), 431–438 Zbl1101.14017
- [47] Goldman W.M., An exposition of results of Fricke and Vogt, preprint available at http://arxiv.org/abs/math/0402103
- [48] Gordeev N., Rehmann U., On multicommutators for simple algebraic groups, J. Algebra, 2001, 245(1), 275–296 Zbl0994.20039
- [49] Gordon S.R., Associators in simple algebras, Pacific J. Math., 1974, 51(1), 131–141 Zbl0348.17010
- [50] Gowers W.T., Quasirandom groups, Combin. Probab. Comput., 2008, 17(3), 363–387 Zbl1191.20016
- [51] Grunewald F., Kunyavskiĭ B., Nikolova D., Plotkin E., Two-variable identities in groups and Lie algebras, J. Math. Sci. (N.Y.), 2003, 116(1), 2972–2981 Zbl1069.20012
- [52] Grunewald F., Kunyavskiĭ B., Plotkin E., Characterization of solvable groups and solvable radical, Internat. J. Algebra Comput., 2013, 23(5), 1011–1062 Zbl1284.20014
- [53] Guralnick R., Malle G., Products of conjugacy classes and fixed point spaces, J. Amer. Math. Soc., 2012, 25(1), 77–121 Zbl1286.20007
- [54] Guralnick R.M., Tiep P.H., Cross characteristic representations of even characteristic symplectic groups, Trans. Amer. Math. Soc., 2004, 356(12), 4969–5023 Zbl1062.20013
- [55] Guralnick R.M., Tiep P.H., The Waring problem for finite quasisimple groups. II, preprint available at http://arxiv.org/abs/1302.0333 Zbl06512679
- [56] Horowitz R.D., Characters of free groups represented in the two-dimensional special linear group, Comm. Pure Appl. Math., 1972, 25(6), 635–649 Zbl1184.20009
- [57] Hrushovski E., The elementary theory of the Frobenius automorphisms, preprint available at http://arxiv.org/abs/math.LO/0406514/
- [58] Humphreys J.E., Modular Representations of Finite Groups of Lie Type, London Math. Soc. Lecture Note Ser., 326, Cambridge University Press, Cambridge, 2006 Zbl1113.20016
- [59] Huppert B., Blackburn N., Finite Groups, III, Grundlehren Math. Wiss., 243, Springer, Berlin-Heidelberg-New York, 1982
- [60] Jambor S., Liebeck M.W., O’Brien E.A., Some word maps that are non-surjective on infinitely many finite simple groups, Bull. Lond. Math. Soc., 2013, 45(5), 907–910 Zbl1292.20014
- [61] Kanel-Belov A., Kunyavskiĭ B., Plotkin E., Word equations in simple groups and polynomial equations in simple algebras, Vestnik St. Petersburg Univ. Math., 2013, 46(1), 3–13 Zbl1300.20033
- [62] Kanel-Belov A., Malev S., Rowen L., The images of non-commutative polynomials evaluated on 2×2 matrices Proc. Amer. Math. Soc., 2012, 140(2), 465–478 Zbl1241.16017
- [63] Kantor W.M., Lubotzky A., The probability of generating a finite classical group, Geom. Dedicata, 1990, 36(1), 67–87 Zbl0718.20011
- [64] Kapovich I., Mapping tori of endomorphisms of free groups, Comm. Algebra, 2000, 28(6), 2895–2917 Zbl0953.20035
- [65] Kapovich I., Schupp P.E., Random quotients of the modular group are rigid and essentially incompressible, J. Reine Angew. Math., 2009, 628, 91–119 Zbl1167.22009
- [66] Kassabov M., Nikolov N., Words with few values in finite simple groups, Quart. J. Math. (in press), DOI: 10.1093/qmath/has018 Zbl1296.20037
- [67] Lang S., Weil A., Number of points of varieties in finite fields, Amer. J. Math., 1954, 76(4), 819–827 Zbl0058.27202
- [68] Larsen M., Word maps have large image, Israel J. Math., 2004, 139, 149–156 Zbl1130.20310
- [69] Larsen M.J., Pink R., Finite subgroups of algebraic groups, J. Amer. Math. Soc., 2011, 24(4), 1105–1158 Zbl1241.20054
- [70] Larsen M., Shalev A., Characters of symmetric groups: sharp bounds and applications, Invent. Math., 2008, 174(3), 645–687 Zbl1166.20009
- [71] Larsen M., Shalev A., Word maps and Waring type problems, J. Amer. Math. Soc., 2009, 22(2), 437–466 Zbl1206.20014
- [72] Larsen M., Shalev A., Fibers of word maps and some applications, J. Algebra, 2012, 354, 36–48 Zbl1258.20011
- [73] Larsen M., Shalev A., Tiep P.H., The Waring problem for finite simple groups, Ann. of Math., 2011, 174(3), 1885–1950 Zbl1283.20008
- [74] Larsen M., Shalev A., Tiep P.H., Waring problem for finite quasisimple groups, Int. Math. Res. Not. (IMRN), 2013, 10, 2323–2348 Zbl1329.20014
- [75] Levy M., Word maps with small image in simple groups, preprint available at http://arxiv.org/abs/1206.1206
- [76] Levy M., Word maps with small image in almost simple groups and quasisimple groups, preprint available at http://arxiv.org/abs/1301.7188
- [77] Lidl R., Mullen G.L., Turnwald G., Dickson Polynomials, Pitman Monogr. Surveys Pure Appl. Math., 65, Longman Scientific & Technical, Harlow, 1993
- [78] Lidl R., Niederreiter H., Finite Fields, Encyclopedia Math. Appl., 20, Addison-Wesley, Reading, 1983
- [79] Liebeck M.W., O’Brien E.A., Shalev A., Tiep P.H., The Ore conjecture, J. Eur. Math. Soc. (JEMS), 2010, 12(4), 939–1008 Zbl1205.20011
- [80] Liebeck M.W., O’Brien E.A., Shalev A., Tiep P.H., Commutators in finite quasisimple groups, Bull. Lond. Math. Soc., 2011, 43(6), 1079–1092 Zbl1236.20011
- [81] Liebeck M.W., O’Brien E.A., Shalev A., Tiep P.H., Products of squares in finite simple groups, Proc. Amer. Math. Soc., 2012, 140(1), 21–33 Zbl1262.20013
- [82] Liebeck M.W., Shalev A., The probability of generating a finite simple group, Geom. Dedicata, 1995, 56(1), 103–113 Zbl0836.20068
- [83] Liebeck M.W., Shalev A., Diameters of finite simple groups: sharp bounds and applications, Ann. of Math., 2001, 154(2), 383–406 Zbl1003.20014
- [84] Liebeck M.W., Shalev A., Fuchsian groups, finite simple groups, and representation varieties, Invent. Math., 2005, 159(2), 317–367 Zbl1134.20059
- [85] Lubotzky A., Images of word maps in finite simple groups, Glasg. Math. J. (in press), DOI:10.1017/S0017089513000396
- [86] Lyndon R.C., Words and infinite permutations, In: Mots, Lang. Raison. Calc., Hermès, Paris, 1990, 143–152
- [87] Macbeath A.M., Generators of the linear fractional groups, In: Number Theory, Houston, 1967, American Mathematical Society, Providence, 1969, 14–32
- [88] Macpherson D., Tent K., Pseudofinite groups with NIP theory and definability in finite simple groups, In: Groups and Model Theory, Mülheim an der Ruhr, May 30–June 3, 2011, Contemp. Math., 576, American Mathematical Society, Providence, 2012, 255–267 Zbl1273.03127
- [89] Magnus W., Rings of Fricke characters and automorphisms groups of free groups, Math. Z., 1980, 170(1), 91–102 Zbl0433.20033
- [90] Magnus W., The uses of 2 by 2 matrices in combinatorial group theory. A survey, Resultate Math., 1981, 4(2), 171–192 Zbl0468.20031
- [91] Manin Yu.I., Cubic Forms, North-Holland Math. Library, 4, North-Holland, Amsterdam, 1986
- [92] Maroli J.A., Representation of tree permutations by words, Proc. Amer. Math. Soc., 1990, 110(4), 859–869 Zbl0746.06008
- [93] Martinez C., Zelmanov E., Products of powers in finite simple groups, Israel J. Math., 1996, 96(2), 469–479 Zbl0890.20013
- [94] Myasnikov A., Nikolaev A., Verbal subgroups of hyperbolic groups have infinite width, preprint available at http://arxiv.org/abs/1107.3719 Zbl06355585
- [95] Myasnikov A.G., Shpilrain V., Automorphic orbits in free groups, J. Algebra, 2003, 269(1), 18–27 Zbl1035.20019
- [96] Najib S., Une généralisation de l’inégalité de Stein-Lorenzini, J. Algebra, 2005, 292(2), 566–573 Zbl1119.13022
- [97] Nikolov N., Algebraic properties of profinite groups, preprint available at http://arxiv.org/abs/1108.5130
- [98] Nikolov N., Pyber L., Product decompositions of quasirandom groups and a Jordan type theorem, J. Eur. Math. Soc. (JEMS), 2011, 13(4), 1063–1077 Zbl1228.20020
- [99] Nikolov N., Segal D., A characterization of finite soluble groups, Bull. Lond. Math. Soc., 2007, 39(2), 209–213 Zbl1122.20007
- [100] Nikolov N., Segal D., Powers in finite groups, Groups Geom. Dyn., 2011, 5(2), 501–507 Zbl1243.20036
- [101] Nikolov N., Segal D., Generators and commutators in finite groups; abstract quotients of compact groups, Invent. Math., 2012, 190(3), 513–602 Zbl1268.20031
- [102] Ore O., Some remarks on commutators, Proc. Amer. Math. Soc., 1951, 2(2), 307–314
- [103] Platonov V.P., Linear groups with identical relations, Dokl. Akad. Nauk BSSR, 1967, 11, 581–582 (in Russian)
- [104] Puder D., Primitive words, free factors and measure preservation, Israel J. Math., 2014 (in press), DOI:10.1007/s11856-013-0055-2 Zbl1308.20023
- [105] Puder D., Parzanchevski O., Measure preserving words are primitive, preprint available at http://arxiv.org/abs/1202.3269 Zbl06394341
- [106] Razmyslov Yu.P., A certain problem of Kaplansky, Math. USSR Izv., 1973, 7(3), 479–496 Zbl0314.16016
- [107] Ribnere E., Sequences of words characterizing finite solvable groups, Monatsh. Math., 2009, 157(4), 387–401 Zbl1195.20016
- [108] Rosset M., Rosset S., Elements of trace zero that are not commutators, Comm. Algebra, 2000, 28(6), 3059–3072 Zbl0954.16021
- [109] Saxl J., Wilson J.S., A note on powers in simple groups, Math. Proc. Cambridge Philos. Soc., 1997, 122(1), 91–94 Zbl0890.20014
- [110] Schul G., Shalev A., Words and mixing times in finite simple groups, Groups Geom. Dyn., 2011, 5(2), 509–527 Zbl1245.20075
- [111] Segal D., Words: Notes on Verbal Width in Groups, London Math. Soc. Lecture Note Ser., 361, Cambridge University Press, Cambridge, 2009 Zbl1198.20001
- [112] Serre J.-P., Le groupe de Cremona et ses sous-groupes finis, In: Séminaire Bourbaki, 2008/2009 (997–1011), Astérisque, 2010, 332(1000), 75–100
- [113] Shalev A., Commutators, words, conjugacy classes and character methods, Turkish J. Math., 2007, 31(Suppl.), 131–148 Zbl1162.20014
- [114] Shalev A., Word maps, conjugacy classes, and a noncommutative Waring-type theorem, Ann. of Math., 2009, 170(3), 1383–1416 Zbl1203.20013
- [115] Shalev A., Applications of some zeta functions in group theory, In: Zeta Functions in Algebra and Geometry, Palma de Mallorca, May 3–7, 2010, Contemp. Math., 566, American Mathematical Society, Providence, 2012, 331–344 Zbl1260.20022
- [116] Slusky M., Zeros of 2×2 matrix polynomials, Comm. Algebra, 2010, 38(11), 4212–4223 Zbl1227.15016
- [117] Suzuki M., On a class of doubly transitive groups, Ann. of Math., 1962, 75(1), 105–145 Zbl0106.24702
- [118] Thom A., Convergent sequences in discrete groups, Canad. Math. Bull., 2013, 56(2), 424–433 Zbl1276.54028
- [119] Thompson J.G., Nonsolvable finite groups all of whose local subgroups are solvable, Bull. Amer. Math. Soc., 1968, 74(3), 383–437 Zbl0159.30804
- [120] Thompson R.C., Commutators in the special and general linear groups, Trans. Amer. Math. Soc., 1961, 101(1), 16–33 Zbl0109.26002
- [121] Tiep P.H., Zalesskii A.E., Some characterizations of the Weil representations of the symplectic and unitary groups, J. Algebra, 1997, 192(1), 130–165 Zbl0877.20030
- [122] Tits J., Free subgroups in linear groups, J. Algebra, 1972, 20(2), 250–270 Zbl0236.20032
- [123] Varshavsky Ya., Lefschetz-Verdier trace formula and a generalization of a theorem of Fujiwara, Geom. Funct. Anal., 2007, 17(1), 271–319 Zbl1131.14019
- [124] Vogt H., Sur les invariants fundamentaux des equations différentielles linéaires du second ordre, Ann. Sci. École Norm. Supér., 1889, 6(Suppl.), 3–70 Zbl21.0314.01
- [125] Wan D., A p-adic lifting and its application to permutation polynomials, In: Finite Fields, Coding Theory, and Advances in Communications and Computing, Las Vegas, August 7–10, 1991, Lecture Notes in Pure and Appl. Math., 141, Marcel Dekker, New York, 1993, 209–216
- [126] Wilson J.S., Characterization of the soluble radical by a sequence of words, J. Algebra, 2011, 326, 286–289 Zbl1243.20029
- [127] Zelmanov E.I., On the restricted Burnside problem, In: Proceedings of the International Congress of Mathematicians, Kyoto, August 21–29, 1990, Mathematical Society of Japan, Tokyo, 1991, 395–402 Zbl0771.20014
- [128] Zorn M., Nilpotency of finite groups, Bull. Amer. Math. Soc., 1936, 42(7), 485–486 Zbl62.0088.10
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.