Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The nilpotency class of finite groups of exponent $ p$


Author: Michael Vaughan-Lee
Journal: Trans. Amer. Math. Soc. 346 (1994), 617-640
MSC: Primary 20D15; Secondary 17B30
DOI: https://doi.org/10.1090/S0002-9947-1994-1264152-8
MathSciNet review: 1264152
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We investigate the properties of Lie algebras of characteristic $ p$ which satisfy the Engel identity $ x{y^n} = 0$ for some $ n < p$. We establish a criterion which (when satisfied) implies that if $ a$ and $ b$ are elements of an Engel-$ n$ Lie algebra $ L$ then $ a{b^{n - 2}}$ generates a nilpotent ideal of $ L$. We show that this criterion is satisfied for $ n = 6,\,p = 7$, and we deduce that if $ G$ is a finite $ m$-generator group of exponent $ 7$ then $ G$ is nilpotent of class at most $ 51{m^8}$.


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

  • [1] S. I. Adjan and A. A. Razborov, Periodic groups and Lie algebras, Uspekhi Mat. Nauk 42 (1987), 3-68. MR 898621 (89i:20033)
  • [2] N. D. Gupta and M. F. Newman, The nilpotency class of finitely generated groups of exponent four, Lecture Notes in Math., vol. 372, Springer-Verlag, Berlin, 1974, pp. 330-332. MR 0352265 (50:4752)
  • [3] P. Hall and G. Higman, On the $ p$-length of $ p$-soluble groups and reduction theorems for Burnside's problem, Proc. London Math. Soc. 6 (1956), 1-42. MR 0072872 (17:344b)
  • [4] 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)
  • [5] G. Havas, G. E. Wall, and J. W. Wamsley, The two generator restricted Burnside group of exponent five, Bull. Austral. Math. Soc. 10 (1974), 459-470. MR 0367056 (51:3298)
  • [6] P. J. Higgins, Lie rings satisfying the Engel condition, Proc. Cambridge Philos. Soc. 50 (1954), 8-15. MR 0059890 (15:596b)
  • [7] G. Higman, On finite groups of exponent five, Proc. Cambridge Philos. Soc. 52 (1956), 381-390. MR 0081285 (18:377e)
  • [8] A. I. Kostrikin, The Burnside problem, Izv. Akad. Nauk SSSR Ser. Mat. 23 (1959), 3-34. MR 0132100 (24:A1947)
  • [9] -, Around Burnside (translated by J. Wiegold), Ergeb. Math. Grenzgeb., Springer-Verlag, Berlin, 1990. MR 1075416 (91i:20038)
  • [10] F. Levi and B. L. van der Waerden, Über eine besondere Klasse von Gruppen, Abh. Math. Sem. Univ. Hamburg 9 (1933), 154-158.
  • [11] Ju. P. Razmyslov, On a problem of Hall-Higman, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), 833-847. MR 508829 (80d:20040)
  • [12] I. N. Sanov, Solution of Burnside's problem for exponent four, Leningrad State Univ. Ann. Math. Ser. 10 (1940), 166-170. MR 0003397 (2:212c)
  • [13] G. Traustason, Engel Lie algebras, Quart. J. Math. (2) 44 (1993), 355-384. MR 1240479 (95a:17015)
  • [14] -, Engel Lie algebras, D. Phil. thesis, Oxford Univ., 1993.
  • [15] M. R. Vaughan-Lee, Lie rings of groups of prime exponent, J. Austral. Math. Soc. 49 (1990), 386-398. MR 1074510 (91m:20059)
  • [16] -, The restricted Burnside problem, Oxford Univ. Press, 1990. MR 1057610 (92c:20001)
  • [17] M. R. Vaughan-Lee and E. I. Zelmanov, Upper bounds in the restricted Burnside problem, J. Algebra 161 (1993). MR 1250531 (94j:20019)
  • [18] E. I. Zelmanov, The solution of the restricted Burnside problem for groups of odd exponent, Izv. Math. USSR 36 (1991), 41-60. MR 1044047 (91i:20037)
  • [19] -, The solution of the restricted Burnside problem for $ 2$-groups, Mat. Sb. 182 (1991), 568-592.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 20D15, 17B30

Retrieve articles in all journals with MSC: 20D15, 17B30


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1264152-8
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society