Book contents
- Frontmatter
- Contents
- Introduction
- Algorithms for matrix groups
- Residual properties of 1-relator groups
- Words and groups
- The modular isomorphism problem for the groups of order 512
- Recent progress in the symmetric generation of groups
- Discriminating groups: a comprehensive overview
- Extending the Kegel Wielandt theorem through π-decomposable groups
- On the prime graph of a finite group
- Applications of Lie rings with finite cyclic grading
- Pronormal subgroups and transitivity of some subgroup properties
- On Engel and positive laws
- Maximal subgroups of odd index in finite groups with simple classical socle
- Some classic and nearly classic problems on varieties of groups
- Generalizations of the Sylow theorem
- Engel groups
- Lie methods in Engel groups
- On the degree of commutativity of p-groups of maximal class
- Class preserving automorphisms of finite p-groups: a survey
- Symmetric colorings of finite groups
- References
Engel groups
Published online by Cambridge University Press: 05 July 2011
Book contents
- Frontmatter
- Contents
- Introduction
- Algorithms for matrix groups
- Residual properties of 1-relator groups
- Words and groups
- The modular isomorphism problem for the groups of order 512
- Recent progress in the symmetric generation of groups
- Discriminating groups: a comprehensive overview
- Extending the Kegel Wielandt theorem through π-decomposable groups
- On the prime graph of a finite group
- Applications of Lie rings with finite cyclic grading
- Pronormal subgroups and transitivity of some subgroup properties
- On Engel and positive laws
- Maximal subgroups of odd index in finite groups with simple classical socle
- Some classic and nearly classic problems on varieties of groups
- Generalizations of the Sylow theorem
- Engel groups
- Lie methods in Engel groups
- On the degree of commutativity of p-groups of maximal class
- Class preserving automorphisms of finite p-groups: a survey
- Symmetric colorings of finite groups
- References
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.
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- Groups St Andrews 2009 in Bath , pp. 520 - 550Publisher: Cambridge University PressPrint publication year: 2011
References
[1] and , On locally finite p-groups satisfying an Engel condition, Proc. Amer. Math. Soc. 130 (2002), 2827–2836.Google Scholar
[2] and , Third Engel groups and the Macdonald–Neumann conjecture, Bull. Austral. Math. Soc. 5 (1971), 379–385.Google Scholar
[4] , and , Groups satisfying semigroup laws and nilpotent-by-Burnside varieties, J. Algebra 195 (1997), 510–525.Google Scholar
[5] and , A note on Engel groups and local nilpotence, J. Austral. Math. Soc. (Series A) 64 (1998), 92–100.Google Scholar
[6] and , Group laws implying virtual nilpotence, J. Austral. Math. Soc. 74 (2003), 295–312.Google Scholar
[7] , On an unsettled question in the theory of discontinous groups, Quart. J. Pure Appl. Math. 37 (1901), 230–238.Google Scholar
[8] , On groups in which every two conjugate operations are permutable, Proc. London Math. Soc. 35 (1902), 28–37.Google Scholar
[9] and , A remark on the structure of n-Engel groups, submitted.
[10] , Une condition suffisante pour qu'n groupe d'Engel soit nilpotent, C. R. Acad. Sci. Paris 308 (1989), 75–78.Google Scholar
[11] , Groupes d'Engel avec la condition maximale sur les p-sous-groupes finis abéliens, Arch. Math. 55 (1990), 313–316.Google Scholar
[12] , On the locally finite p-groups in certain varieties of groups, Quart. J. Math. 48 (1997), 169–178.Google Scholar
[13] , Bounds for nilpotent-by-finite groups in certain varieties, J. Austral. Math. Soc. 73 (2002), 393–404.Google Scholar
[14] and , On varieties in which soluble groups are torsion-by-nilpotent, Internat. J. Algebra Comput. 15 (2005), 537–545.Google Scholar
[15] and , Linear groups with Engel's condition, Dokl. Akad. Nauk BSSR 6 (1962), 277–280.Google Scholar
[16] , Some problems of Burnside type, Internat. Congress Math. Moscow (1966), 284–298 =Amer. Math. Soc. Translations (2) 84 (1969), 83–88.Google Scholar
[19] , The upper central series in soluble groups, Illinois J. Math. 5 (1961), 436–466.Google Scholar
[20] , Varieties of soluble groups and a dichotomy of P. Hall, Bull Austral. Math. Soc. 5 (1971), 394–410.Google Scholar
[27] and , 4-Engel groups are locally nilpotent, Internat. J. Algebra Comput. 15 (2005), 649–682.Google Scholar
[29] , Finite groups in which conjuate operations are commutative, Amer. J. Math. 51, (1929), 35–41Google Scholar
[31] and , Orderable groups satisfying an Engel condition, in Ordered algebraic structures (Gainesville, FL, 1991), 73–79, Kluwer Acad. Publ., Dordrecht, 1993.Google Scholar
[32] and , Weak maximality condition and polycyclic groups, Proc. Amer. Math. Soc. 123, (1995), 711–714.Google Scholar
[33] , Groups in which the commutator operations satisfies certain algebraic conditions, J. Indian Math. Soc. 6 (1942), 87–97.Google Scholar
[34] and , 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] and , On some classes of orderable groups, Rend. Sem. Mat. Fis. Milano 68 (2001), 203–216.Google Scholar
[37] and (eds.), The Kourovka Notebook, Sixteenth Edition, Russian Academy of Sciences, Siberian Division Novosibirsk 2006, Unsolved problems in group theory.Google Scholar
[39] , On right-Engel elements of length three, Proc. Royal Irish Academy 96A (1996), 17–24.Google Scholar
[40] and , Engel 4 groups of exponent 5. II. Orders, Proc. London Math. Soc. (3) 79 (1999), 283–317.Google Scholar
[41] , Computation of nilpotent Engel groups, J. Austral. Math. Soc. 67 (1999), 214–222.Google Scholar
[42] , Engel elements of groups with maximal condition on abelian subgroups, Nanta Math. 1 (1966), 23–28.Google Scholar
[45] , Combinatorial conditions in residually finite groups II, J. Algebra 157 (1993), 51–62.Google Scholar
[46] , Bounded Engel groups with all subgroups subnormal, Comm. Algebra 30 (2002), no. 2, 907–909.Google Scholar
[50] , Locally nilpotent 4-Engel groups are Fitting groups, J. Algebra 270 (2003), 7–27.Google Scholar
[53] , A note on the local nilpotence of 4-Engel groups, Internat. J. Algebra Comput. 15 (2005), 757–764.Google Scholar
[56] , Two-generator conditions for residually finite groups, Bull. London Math. Soc. 23 (1991), 239–248.Google Scholar
[57] and , Identities for Lie algebras of pro-p groups, J. Pure Appl. Algebra 81 (1992), 103–109.Google Scholar
[59] , Some problems in the theory of groups and Lie algebras, Mat. Sb. 180 (1989), 159–167.Google Scholar
[60] , The solution of the restricted Burnside problem for groups of odd exponent, Math. USSR Izv. 36 (1991), 41–60.Google Scholar
[61] , The solution of the restricted Burnside problem for 2-groups, Mat. Sb. 182 (1991), 568–592.Google Scholar
[62] , On additional laws in the Burnside problem on periodic groups, Internat. J. Algebra Comput. 3 (1993), 583–600.Google Scholar
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