Book Review
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MathSciNet review:
1568011
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Book Information:
Author:
A.\ I.\ Kostrikin
Title:
Around Burnside
Additional book information:
Ergeb. Math. Grenzgeb. (3), vol. 20, Springer-Verlag, New York, Berlin, and Heidelberg, 1990, 220 pp., US$82.00. ISBN 0-387-50602-0.
Author:
Michael Vaughan-Lee
Title:
The restricted Burnside problem
Additional book information:
London Math. Soc. Monographs (N.S.), vol. 5, Oxford University Press, Oxford, 1990, 209 pp., US$57.50, ISBN 0-19-853573-2.
S. I. Adyan, Problema Bernsaĭ da i tozhdestva v gruppakh, Izdat. “Nauka”, Moscow, 1975 (Russian). MR 0432770
S. I. Adyan and A. A. Razborov, Periodic groups and Lie algebras, Uspekhi Mat. Nauk 42 (1987), no. 2(254), 3–68, 287 (Russian). MR 898621
S. I. Adyan and N. N. Repin, Lower bounds on the order of maximal periodic groups of prime period, Mat. Zametki 44 (1988), no. 2, 161–176, 286 (Russian); English transl., Math. Notes 44 (1988), no. 1-2, 571–579 (1989). MR 969265, DOI 10.1007/BF01159250
Michael Aschbacher, Peter B. Kleidman, and Martin W. Liebeck, Exponents of almost simple groups and an application to the restricted Burnside problem, Math. Z. 208 (1991), no. 3, 401–409. MR 1134585, DOI 10.1007/BF02571536
Seymour Bachmuth, Horace Y. Mochizuki, and David Walkup, A nonsolvable group of exponent $5$, Bull. Amer. Math. Soc. 76 (1970), 638–640. MR 257209, DOI 10.1090/S0002-9904-1970-12469-7
[6] W. Burnside, On an unsettled question in the theory of discontinuous groups, Quart. J. Pure Appl. Math. 33 (1902), 230-238.
Bruce Chandler and Wilhelm Magnus, The history of combinatorial group theory, Studies in the History of Mathematics and Physical Sciences, vol. 9, Springer-Verlag, New York, 1982. A case study in the history of ideas. MR 680777, DOI 10.1007/978-1-4613-9487-7
E. S. Golod, On nil-algebras and finitely approximable $p$-groups, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 273–276 (Russian). MR 0161878
N. D. Gupta and M. F. Newman, The nilpotency class of finitely generated groups of exponent four, Proceedings of the Second International Conference on the Theory of Groups (Australian Nat. Univ., Canberra, 1973) Lecture Notes in Math., Vol. 372, Springer, Berlin, 1974, pp. 330–332. MR 0352265
P. Hall and Graham Higman, On the $p$-length of $p$-soluble groups and reduction theorems for Burnside’s problem, Proc. London Math. Soc. (3) 6 (1956), 1–42. MR 72872, DOI 10.1112/plms/s3-6.1.1
George Havas, M. F. Newman, and M. R. Vaughan-Lee, A nilpotent quotient algorithm for graded Lie rings, J. Symbolic Comput. 9 (1990), no. 5-6, 653–664. Computational group theory, Part 1. MR 1075429, DOI 10.1016/S0747-7171(08)80080-6
Graham Higman, On finite groups of exponent five, Proc. Cambridge Philos. Soc. 52 (1956), 381–390. MR 81285
A. I. Kostrikin, On Burnside’s problem, Dokl. Akad. Nauk SSSR 119 (1958), 1081–1084 (Russian). MR 0132101
A. I. Kostrikin, The Burnside problem, Izv. Akad. Nauk SSSR Ser. Mat. 23 (1959), 3–34 (Russian). MR 0132100
A. I. Kostrikin, Sandwiches in Lie algebras, Mat. Sb. (N.S.) 110(152) (1979), no. 1, 3–12, 159 (Russian). MR 548513
[16] F. Levi and B. L. van der Waerden, Über eine besondere Klasse von Gruppen, Abh. Math. Sem. Univ. Hamburg 9 (1933), 154-158.
Wilhelm Magnus, A connection between the Baker-Hausdorff formula and a problem of Burnside, Ann. of Math. (2) 52 (1950), 111–126. MR 38964, DOI 10.2307/1969512
Ju. P. Razmyslov, Lie algebras satisfying Engel conditions, Algebra i Logika 10 (1971), 33–44 (Russian). MR 0294428
Ju. P. Razmyslov, The Hall-Higman problem, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), no. 4, 833–847 (Russian). MR 508829
M. R. Vaughan-Lee, Lie rings of groups of prime exponent, J. Austral. Math. Soc. Ser. A 49 (1990), no. 3, 386–398. MR 1074510
E. I. Zel′manov, Solution of the restricted Burnside problem for groups of odd exponent, Izv. Akad. Nauk SSSR Ser. Mat. 54 (1990), no. 1, 42–59, 221 (Russian); English transl., Math. USSR-Izv. 36 (1991), no. 1, 41–60. MR 1044047
E. I. Zel′manov, Solution of the restricted Burnside problem for $2$-groups, Mat. Sb. 182 (1991), no. 4, 568–592 (Russian); English transl., Math. USSR-Sb. 72 (1992), no. 2, 543–565. MR 1119009
E. I. Zel′manov and A. I. Kostrikin, A theorem on sandwich algebras, Trudy Mat. Inst. Steklov. 183 (1990), 106–111, 225 (Russian). Translated in Proc. Steklov Inst. Math. 1991, no. 4, 121–126; Galois theory, rings, algebraic groups and their applications (Russian). MR 1092020
- [1]
- S. I. Adyan, The Burnside problem and identities in groups, Izdat. "Nauka", Moscow, 1975; English transl., Ergebnisse der Math. Bd. 95, Springer-Verlag, Berlin, 1979. MR 0432770 (55:5753)
- [2]
- S. I. Adyan and A. A. Razborov, Periodic groups and Lie algebras, Uspekhi Mat. Nauk 42 (1987), 3-68. (Russian) MR 898621 (89i:20033)
- [3]
- S. I. Adyan and N. N. Repin, Lower bounds for the orders of maximal groups of prime exponent, Mat. Zametki 44 (1988), 161-176. (Russian) MR 969265 (90f:20054)
- [4]
- M. Aschbacher, P. B. Kleidman, and M. W. Liebeck, Exponents of almost simple groups and an application to the restricted Burnside problem, Math. Z. 208 (1991), 401-409. MR 1134585 (92k:20058)
- [5]
- S. Bachmuth, H. Y. Mochizuki, and D. Walkup, A nonsolvable group of exponent 5, Bull. Amer. Math. Soc. 76 (1970), 638-640. MR 0257209 (41:1862)
- [6]
- W. Burnside, On an unsettled question in the theory of discontinuous groups, Quart. J. Pure Appl. Math. 33 (1902), 230-238.
- [7]
- B. Chandler and W. Magnus, The history of combinatorial group theory: a case study in the history of ideas, Springer-Verlag, New York, 1982. MR 680777 (85c:01001)
- [8]
- E. S. Golod, On nil-algebras and finitely approximable p-groups, Akad. Nauk SSSR Ser. Mat. 28 (1964), 273-276; English transl., Amer. Math. Soc. Transl. Ser. 2 48 (1965), 103-106. MR 0161878 (28:5082)
- [9]
- N. D. Gupta and M. F. Newman, The nilpotency class of finitely generated groups of exponent 4, Lecture Notes in Math., vol. 372, Springer-Verlag, Berlin and New York, 1974, pp. 330-332. MR 0352265 (50:4752)
- [10]
- P. Hall and G. Higman, On the p-length of p-soluble groups and reduction theorems for Burnside's problem, Proc. London Math. Soc. (3) 6 (1956), 1-42. MR 0072872 (17:344b)
- [11]
- G. Havas, M. F. Newman and M. R. Vaughan-Lee, A nilpotent quotient algorithm for graded Lie rings, J. Symbolic Comput. 9 (1990), 653-664. MR 1075429 (92d:20054)
- [12]
- G. Higman, On finite groups of exponent 5, Proc. Cambridge Philos. Soc. 52 (1956), 381-390. MR 0081285 (18:377e)
- [13]
- A. I. Kostrikin, On Burnside's problem, Dokl. Akad. Nauk SSSR 119 (1958), 1081-1084. (Russian) MR 0132101 (24:A1948)
- [14]
- -, On Burnside's problem, Izv. Akad. Nauk SSSR Ser. Math. 23 (1959), 3-34. (Russian) MR 0132100 (24:A1947)
- [15]
- -, Sandwiches in Lie algebras, Mat. Sb. 110 (1979), 3-12. (Russian) MR 548513 (81e:17004)
- [16]
- F. Levi and B. L. van der Waerden, Über eine besondere Klasse von Gruppen, Abh. Math. Sem. Univ. Hamburg 9 (1933), 154-158.
- [17]
- W. Magnus, A connection between the Baker-Hausdorff formula and a problem of Burnside, Ann. of Math. 52 (1950), 111-126; Errata, Ann. of Math. 57 (1953), 606. MR 0038964 (12:476c)
- [18]
- Yu. P. Razmyslov, On Engel Lie algebras, Algebra i Logika 10 (1971), 33-44. (Russian) MR 0294428 (45:3498)
- [19]
- -, On a problem of Hall and Higman, Izv. Akad. Nauk SSSR, Ser. Mat. 42 (1978), 833-847. (Russian) MR 508829 (80d:20040)
- [20]
- M. R. Vaughan-Lee, Lie rings of groups of prime exponent, J. Austral. Math. Soc. Ser. A 49 (1990), 386-398. MR 1074510 (91m:20059)
- [21]
- E. I. Zel'manov, Solution of the restricted Burnside problem for groups of odd exponent, Izv. Akad. Nauk SSSR, Ser. Mat. 54 (1990), 42-59. (Russian) MR 1044047 (91i:20037)
- [22]
- -, Solution of the restricted Burnside problem for 2-groups, Mat. Sb. 182 (1991), 568-592. (Russian) MR 1119009 (93a:20063)
- [23]
- E. I. Zel'manov and A. I. Kostrikin, A theorem on sandwich algebras, Trudy Mat. Inst. Steklov 183 (1990), 106-111. (Russian) MR 1092020 (92h:17007)
Review Information:
Reviewer:
M. F. Newman
Reviewer:
G. E. Wall
Journal:
Bull. Amer. Math. Soc.
28 (1993), 157-161
DOI:
https://doi.org/10.1090/S0273-0979-1993-00341-0
