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Engel groups

Published online by Cambridge University Press:  05 July 2011

By
Gunnar Traustason
Edited by
C. M. Campbell ,
M. R. Quick ,
E. F. Robertson ,
C. M. Roney-Dougal ,
G. C. Smith  and
G. Traustason
Gunnar Traustason
Affiliation:
University of Bath
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
G. Traustason
Affiliation:
University of Bath
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Summary

Abstract

We give a survey on Engel groups with particular emphasis on the development during the last two decades.

Introduction

We define the n-Engel word en(x, y) as follows: e0(x, y) = x and en+1(x, y) = [en(x, y), y]. We say that a group G is an Engel group if for each pair of elements a, b ∈ G we have en(a, b) = 1 for some positive integer n = n(a, b). If n can be chosen independently of a, b then G is an n-Engel group.

One can also talk about Engel elements. An element a ∈ G is said to be a left Engel element if for all g ∈ G there exists a positive integer n = n(g) such that en(g, a) = 1. If instead one can for all g ∈ G choose n = n(g) such that en(a, g) = 1 then a is said to be a right Engel element. If in either case we can choose n independently of g then we talk about left n-Engel or right n-Engel element respectively.

So to say that a is left 1-Engel or right 1-Engel element is the same as saying that a is in the center and a group G is 1-Engel if and only if G is abelian. Every group that is locally nilpotent is an Engel group. Furthermore for any group G we have that all the elements of the locally nilpotent radical are left Engel elements and all the elements in the hyper-center are right Engel elements.

Type
Chapter
Information
Groups St Andrews 2009 in Bath , pp. 520 - 550
Publisher: Cambridge University Press
Print publication year: 2011

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References

[1] A., Abdollahi and G., Traustason, On locally finite p-groups satisfying an Engel condition, Proc. Amer. Math. Soc. 130 (2002), 2827–2836.Google Scholar
[2] S., Bachmuth and H. Y., Mochizuki, Third Engel groups and the Macdonald–Neumann conjecture, Bull. Austral. Math. Soc. 5 (1971), 379–385.Google Scholar
[3] R., Baer, Engelsche Elemente Noetherscher Gruppen, Math. Ann. 133 (1957), 256–270.Google Scholar
[4] R. G., Burns, O., Madcedoćnska and Y., Medvedev, Groups satisfying semigroup laws and nilpotent-by-Burnside varieties, J. Algebra 195 (1997), 510–525.Google Scholar
[5] R. G., Burns and Y., Medvedev, A note on Engel groups and local nilpotence, J. Austral. Math. Soc. (Series A) 64 (1998), 92–100.Google Scholar
[6] R. G., Burns and Y., Medvedev, Group laws implying virtual nilpotence, J. Austral. Math. Soc. 74 (2003), 295–312.Google Scholar
[7] W., Burnside, On an unsettled question in the theory of discontinous groups, Quart. J. Pure Appl. Math. 37 (1901), 230–238.Google Scholar
[8] W., Burnside, On groups in which every two conjugate operations are permutable, Proc. London Math. Soc. 35 (1902), 28–37.Google Scholar
[9] P. G., Crosby and G., Traustason, A remark on the structure of n-Engel groups, submitted.
[10] G., Endimioni, Une condition suffisante pour qu'n groupe d'Engel soit nilpotent, C. R. Acad. Sci. Paris 308 (1989), 75–78.Google Scholar
[11] G., Endimioni, Groupes d'Engel avec la condition maximale sur les p-sous-groupes finis abéliens, Arch. Math. 55 (1990), 313–316.Google Scholar
[12] G., Endimioni, On the locally finite p-groups in certain varieties of groups, Quart. J. Math. 48 (1997), 169–178.Google Scholar
[13] G., Endimioni, Bounds for nilpotent-by-finite groups in certain varieties, J. Austral. Math. Soc. 73 (2002), 393–404.Google Scholar
[14] G., Endimioni and G., Traustason, On varieties in which soluble groups are torsion-by-nilpotent, Internat. J. Algebra Comput. 15 (2005), 537–545.Google Scholar
[15] M. S., Garascuk and D. A., Suprunenko, Linear groups with Engel's condition, Dokl. Akad. Nauk BSSR 6 (1962), 277–280.Google Scholar
[16] E. S., Golod, Some problems of Burnside type, Internat. Congress Math. Moscow (1966), 284–298 =Amer. Math. Soc. Translations (2) 84 (1969), 83–88.Google Scholar
[17] M. I., Golovanov, Nilpotency class of 4-Engel Lie rings, Algebra Logika 25 (1986), 508–532.Google Scholar
[18] K. W., Gruenberg, Two theorems on Engel groups, Proc. Cambridge Philos. Soc. 49 (1953), 377–380.Google Scholar
[19] K. W., Gruenberg, The upper central series in soluble groups, Illinois J. Math. 5 (1961), 436–466.Google Scholar
[20] J. R. J., Groves, Varieties of soluble groups and a dichotomy of P. Hall, Bull Austral. Math. Soc. 5 (1971), 394–410.Google Scholar
[21] N. D., Gupta, Third-Engel 2-groups are soluble, Canad. Math. Bull. 15 (1972), 523–524.Google Scholar
[22] N. D., Gupta and F., Levin, On soluble Engel groups and Lie algebras, Arch. Math. 34 (1980), 289–295.Google Scholar
[23] N. D., Gupta and M. F., Newman, On metabelian groups, J. Austral. Math. Soc. 6 (1966), 362–368.Google Scholar
[24] N. D., Gupta and M. F., Newman, Third Engel groups, Bull. Austral. Math. Soc. 40 (1989), 215–230.Google Scholar
[25] N. D., Gupta and K. W., Weston, On groups of exponent four, J. Algebra 17 (1971), 59–66.Google Scholar
[26] P., Hall, Some word-problems, J. London Math. Soc. 33 (1958), 482–496.Google Scholar
[27] G., Havas and M. R., Vaughan-Lee, 4-Engel groups are locally nilpotent, Internat. J. Algebra Comput. 15 (2005), 649–682.Google Scholar
[28] H., Heineken, Engelsche Elemente der Länge drei, Illinois J. Math. 5 (1961), 681–707.Google Scholar
[29] C., Hopkins, Finite groups in which conjuate operations are commutative, Amer. J. Math. 51, (1929), 35–41Google Scholar
[30] L. C., Kappe and W. P., Kappe, On three-Engel groups, Bull. Austral. Math. Soc. 7 (1972), 391–405.Google Scholar
[31] Y. K., Kim and A. H., Rhemtulla, Orderable groups satisfying an Engel condition, in Ordered algebraic structures (Gainesville, FL, 1991), 73–79, Kluwer Acad. Publ., Dordrecht, 1993.Google Scholar
[32] Y. K., Kim and A. H., Rhemtulla, Weak maximality condition and polycyclic groups, Proc. Amer. Math. Soc. 123, (1995), 711–714.Google Scholar
[33] F. W., Levi, Groups in which the commutator operations satisfies certain algebraic conditions, J. Indian Math. Soc. 6 (1942), 87–97.Google Scholar
[34] P., Longobardi and M., Maj, Semigroup identities and Engel groups, in Groups St Andrews 1997 in Bath II, 527–531, London Math. Soc. Lecture Note Ser. 261, Cambridge Univ. Press, Cambridge 1999.Google Scholar
[35] P., Longobardi and M., Maj, On some classes of orderable groups, Rend. Sem. Mat. Fis. Milano 68 (2001), 203–216.Google Scholar
[36] I. D., Macdonald and B. H., Neumann, A third-Engel 5-group, J. Austral. Math. Soc. 7 (1967), 555–569.Google Scholar
[37] V. D., Mazurov and E. I., Khukhro (eds.), The Kourovka Notebook, Sixteenth Edition, Russian Academy of Sciences, Siberian Division Novosibirsk 2006, Unsolved problems in group theory.Google Scholar
[38] Y., Medvedev, On compact Engel groups, Israel J. Math. 135 (2003), 147–156.Google Scholar
[39] M., Newell, On right-Engel elements of length three, Proc. Royal Irish Academy 96A (1996), 17–24.Google Scholar
[40] M. F., Newman and M. R., Vaughan-Lee, Engel 4 groups of exponent 5. II. Orders, Proc. London Math. Soc. (3) 79 (1999), 283–317.Google Scholar
[41] W., Nickel, Computation of nilpotent Engel groups, J. Austral. Math. Soc. 67 (1999), 214–222.Google Scholar
[42] T. A., Peng, Engel elements of groups with maximal condition on abelian subgroups, Nanta Math. 1 (1966), 23–28.Google Scholar
[43] F., Point, Milnor identities, Comm. Algebra 24 (1996), 3725–3744.Google Scholar
[44] Ju. P., Razmyslov, The Hall-Higman problem, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), 833–847.Google Scholar
[45] A., Shalev, Combinatorial conditions in residually finite groups II, J. Algebra 157 (1993), 51–62.Google Scholar
[46] H., Smith, Bounded Engel groups with all subgroups subnormal, Comm. Algebra 30 (2002), no. 2, 907–909.Google Scholar
[47] G., Traustason, Engel Lie-algebras, Quart. J. Math. Oxford (2) 44 (1993), 355–384.Google Scholar
[48] G., Traustason, On 4-Engel groups, J. Algebra 178 (1995), 414–429.Google Scholar
[49] G., Traustason, Semigroup identities in 4-Engel groups, J. Group Theory 2 (1999), 39–46.Google Scholar
[50] G., Traustason, Locally nilpotent 4-Engel groups are Fitting groups, J. Algebra 270 (2003), 7–27.Google Scholar
[51] G., Traustason, Milnor groups and (virtual) nilpotence, J. Group Theory 8 (2005), 203–221.Google Scholar
[52] G., Traustason, Two generator 4-Engel groups, Internat. J. Algebra Comput. 15 (2005), 309–316.Google Scholar
[53] G., Traustason, A note on the local nilpotence of 4-Engel groups, Internat. J. Algebra Comput. 15 (2005), 757–764.Google Scholar
[54] M. R., Vaughan-Lee, Engel-4 groups of exponent 5, Proc. London Math. Soc. (3) 74 (1997), 306–334.Google Scholar
[55] M. R., Vaughan-Lee, On 4-Engel groups, LMS J. Comput. Math. 10 (2007), 341–353.Google Scholar
[56] J. S., Wilson, Two-generator conditions for residually finite groups, Bull. London Math. Soc. 23 (1991), 239–248.Google Scholar
[57] J. S., Wilson and E. I., Zel'manov, Identities for Lie algebras of pro-p groups, J. Pure Appl. Algebra 81 (1992), 103–109.Google Scholar
[58] E. I., Zel'manov, Engel Lie-algebras, Dokl. Akad. Nauk SSSR 292 (1987), 265–268.Google Scholar
[59] E. I., Zel'manov, Some problems in the theory of groups and Lie algebras, Mat. Sb. 180 (1989), 159–167.Google Scholar
[60] E. I., Zel'manov, The solution of the restricted Burnside problem for groups of odd exponent, Math. USSR Izv. 36 (1991), 41–60.Google Scholar
[61] E. I., Zel'manov, The solution of the restricted Burnside problem for 2-groups, Mat. Sb. 182 (1991), 568–592.Google Scholar
[62] E. I., Zel'manov, On additional laws in the Burnside problem on periodic groups, Internat. J. Algebra Comput. 3 (1993), 583–600.Google Scholar
[63] M., Zorn, Nilpotency of finite groups, Bull. Amer. Math. Soc. 42 (1936), 485–486.Google Scholar

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