Abstract
The purpose of this chapter is to describe the use of Lie-theoretic methods in the study of pro-p Groupss. I shall also discuss briefly some related objects, such as finite p-groups and residually finite groups. Aspects of this topic feature in several books and survey papers; see for instance [62], [45] Chapter VIII, [42], [137], [27], [57], [58], [148], [121]. In this survey I will try to focus on the most recent developments and applications, which are mostly not covered in the sources mentioned above.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A.G. Abercrombie, Subgroups and subrings of profinite rings, Math. Proc. Cambr. Phil. Soc., 116 (1994), 209–222.
A.A. Albert and M.S. Frank, Simple Lie algebras of characteristic p, Rend. Torino, 14 (1954/5), 117–139.
J.L. Alperin, Automorphisms of solvable groups, Proc. Amer. Math. Soc., 13 (1962), 175–180.
J.L. Alperin and G. Glauberman, Limits of abelian subgroups of finite p-groups, J. Alg., 203 (1998), 533–566.
M. Artin and J.T. Stafford, Noncommutative graded domains with quadratic growth,Invent. Math., 122 (1995), 231–276.
M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math., 76 (1984), 469–514.
Y.A. Bahturin and M.V. Zaicev, Identities of graded algebras, J. Alg., 205 (1998), 1–12.
Y. Barnea, The lower rank of some pro-p Groupss and the number of generators of simple Lie algebras. Preprint, 1998.
Y. Barnea and B. Klopsch, Index subgroups of the Nottingham group. Preprint, 1999.
Y. Barnea and M. Larsen, A non-abelian free pro-p group is not linear over a local field, J. Alg., 214 (1999), 338–341.
Y. Barnea and M. Larsen, Random generation in simple algebraic groups over local fields. Preprint, 1998.
Y. Barnea and A. Shalev, Hausdorff dimension, Pro-p groups, and Kac-Moody algebras, Trans. Amer. Math. Soc., 349 (1997), 5073–5091.
Y. Barnea, A. Shalev and E.I. Zelmanov, Graded subalgebras of affine KacMoody algebras, Israel J. Math., 104 (1998), 321–334.
G. Benkart, A.I. Kostrikin and M.I. Kuznetsov, Finite-dimensional Lie algebras with a nonsingular derivation, J. Alg., 171 (1995), 894–916.
Y. Benoist, Une nilvariété non affine, C.R. Acad. Sci. Paris, 315 (1992), 983–986.
G.M. Bergman, A note on growth functions of algebras and semigroups, mimeographed notes, University of California, Berkeley, 1978.
N. Blackburn, On a special class of p-groups, Acta Math., 100 (1958), 49–92.
D. Burde and F. Grunewald, Modules for certain Lie algebras of maximal class, J. PureAppl. Alg., 99 (1995), 239–254.
R.G. Burns, O. Macedoñska and Yu. Medvedev, Groups satisfying semigroup laws, and nilpotent-by-Burnside varieties, J. Alg., 195 (1997), 510–525.
R.G. Burns and Yu. Medvedev, A note on Engel groups and local nilpotence, J. Austral. Math. Soc. (Ser. A), 64 (1998), 92–100.
R. Camina, Subgroups of the Nottingham group, J. Alg., 196 (1997), 101–113.
A. Caranti, S. Mattarei and M.F. Newman, Graded Lie algebras of maximal class, Trans. Amer. Math. Soc., 349 (1997), 4021–4051.
A. Caranti and M.F. Newman, Graded Lie algebras of maximal class. II, to appear.
C. Chevalley, Théorie des groupes de Lie: groupes algébriques, Théorèmes généraux sur les algèbres de Lie, Hermann, Paris.
J.D. Dixon, The probability of generating the symmetric group, Math. Z., 110 (1969), 199–205.
J.D. Dixon, L. Pyber, A. Seress and A. Shalev, Residual properties of free groups and probabilistic methods. Preprint, 1999.
J.D. Dixon, M.P.F. du Sautoy, A. Mann and D. Segal, Analytic pro-p Groupss, 2nd Edition, Cambridge University Press, Cambridge, U.K., 1999.
S. Donkin, Space groups and groups of prime power order. VIII. Pro-p groups of finite coclass and p-adic Lie algebras, J. Alg., 111 (1987), 316–342.
E.B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Transi., 6 (2) (1957), 111–244.
M.P.F. du Sautoy, Finitely generated groups, p-adic analytic groups, and Poincaré series, Ann. Math., 137 (1993), 639–670.
M.P.F. du Sautoy, Pro-p groups, in CRM Proc. and Lecture Notes 17 (1999) (Banff Summer School Proceedings), pp. 99–130.
M.P.F. du Sautoy, p-Groups, coclass and model theory: a proof of conjecture P. Preprint, 1999.
E.B. Dynkin, Semisimple subalgebras of semisimple Lie algebras, Amer. Math. Soc. Transi., 6 (2) (1957), 111–244.
E.B. Dynkin, Maximal subgroups of the classical groups, Amer. Math. Soc. Transi., 6 (2) (1957), 245–378.
K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, New York, 1990.
A. Fialowski, Classification of graded Lie algebras with two generators, Moscow Univ. Math. Bull., 38 (1983), 76–79.
I. M. Gelfand and A. A. Kirillov, Sur les corps liés aux algèbres enveloppantes des algèbres de Lie, Publ. Math. IHES, 31 (1966), 509–523.
G. Glauberman, Large abelian subgroups of finite p-groups, J. Alg., 196 (1997), 301–338.
E.S. Golod, On nil algebras and finitely approximable groups, Izv. AN SSSR, 28 (1964), 273–276. (in Russian)
K.W. Gruenberg, Two theorems on Engel groups, Proc. Cambr. Phil. Soc., 49 (1953), 377–380.
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.
B. Hartley, Topics in the Theory of Nilpotent Groups, in: Group Theory —Essays for Philip Hall, eds: K.W. Gruenberg and J.E. Roseblade, Academic Press, London, 1984.
M. Hazewinkel, Formal Groups and Applications, Academic Press, New York, 1978.
G. Higman, Groups and Lie rings having automorphisms without non-trivial fixed points, J. London Math. Soc., 32 (1957), 321–334.
B. Huppert and N. Blackburn, Finite Groups, II, Springer, Berlin, 1982.
I. Ilani, Counting finite index subgroups and the P. Hall enumeration principle, Israel J. Math., 68 (1989), 18–26.
I. Ilani, Analytic pro-p groups and their Lie algebras, J. Alg., 176 (1995), 34–58.
N. Jacobson, Lie Algebras, Wiley-Interscience, New York, 1962.
V.G. Kac, Simple graded Lie algebras of finite growth, Math. USSR Izv., 2 (1968), 1271–1311.
V.G. Kac, Infinite Dimensional Lie Algebras, Progress in Mathematics 44, Birkhäuser, Boston, 1983.
W.M. Kantor and A. Lubotzky, The probability of generating a finite classical group, Geom. Ded., 36 (1990), 67–87.
V. Kharchenko, Galois extensions and rings of quotients, Algebra i Logika, 13 (4) (1974), 460–484.
E.I. Khukhro, Locally nilpotent groups admitting a splitting automorphism of prime order, Mat. Sbornik, 130 (1986), 120–127. English Transi.: Math. USSR Sbornik,58 (1987), 119–126.
E.I. Khukhro, On the Hughes problem for finite p-groups, Algebra i Logika, 26 (1987), 642–646. English Transi.: Algebra and Logic,26 (1988), 398–401.
E.I. Khukhro, Groups and Lie rings admitting an almost regular automorphism of prime order, Mat. Sbornik, 181 (1990), 1207–1219. English transi.: Math. USSR Sbornik, 71 (1992), 51–63.
E.I. Khukhro, Finite p-groups admitting p-automorphisms with few fixed points, Mat. Sbornik, 184 (1993), 53–64. English transl.: Russian Acad. Sci. Sbornik Math., 80 (1995), 435–444.
E.I. Khukhro, Nilpotent Groups and their Automorphisms, de Gruyter, Berlin, 1993.
E.I. Khukhro, p-Automorphisms of Finite p-groups, London Math. Soc. Lecture Note Series 246, Cambridge University Press, Cambridge, 1998.
I. Kiming, Structure and derived length of finite p-groups possessing an automorphism of p-power order having exactly p fixed points, Math. Scand.,62 (1998), 153–172.
G. Klaas, C.R. Leedham-Green and W. Plesken, Linear pro-p Groupss of Finite Width, Springer Lecture Note Series 1674, Berlin, 1997.
A.I. Kostrikin, On the Burnside problem, Izv. AN SSSR, 23 (1959), 3–34. (in Russian)
A.I. Kostrikin, Around Burnside, Nauka, Moscow, 1986. English Transi.: Springer, Berlin, 1990.
A.I. Kostrikin and I.R. Shafarevich, Graded Lie algebras of finite characteristic, Math. USSR-Izv,3 (1969), 237–304.
G.R. Krause and T.H. Lenagan, Growth of Algebras and Gelfand—Kirillov Dimension, Pitman, London, 1985.
V.A. Kreknin, Solvability of Lie algebras with a regular automorphism of finite period, Soviet. Math. Dokl., 4 (1963), 683–685.
V.A. Kreknin and A.I. Kostrikin, Lie algebras with a regular automorphism, Soviet. Math. Dokl., 4 (1963), 355–358.
M. Lazard, Sur les groupes nilpotents et les anneaux de Lie, Ann. Sci. École Norm. Sup., 71 (1954), 101–190.
M. Lazard, Groupes analytiques p-adiques, Publ. Math. I.H.E.S., 26 (1965), 389–603.
C.R. Leedham-Green, Pro-p groups of finite coclass, J. London Math. Soc., 50 (1994), 43–48.
C.R. Leedham-Green, The structure of finite p-groups, J. London Math. Soc., 50 (1994), 49–67.
C.R. Leedham-Green and S. McKay, On p-groups of maximal class I, Quart. J. Math. Oxford, 27 (1976), 297–311.
C.R. Leedham-Green and S. McKay, On p-groups of maximal class II, Quart. J. Math. Oxford, 29 (1978), 175–186.
C.R. Leedham-Green and S. McKay, On p-groups of maximal class III, Quart. J. Math. Oxford, 29 (1978), 281–299.
C.R. Leedham-Green, S. McKay and W. Plesken, Space groups and groups of prime power order. V. A bound to the dimension of space groups with fixed coclass, Proc. London Math. Soc., 52 (1986), 73–94.
C.R. Leedham-Green, S. McKay and W. Plesken, Space groups and groups of prime power order. VI. A bound to the dimension of a 2-adic group with fixed coclass, J. London Math. Soc., 34 (1986), 417–425.
C.R. Leedham-Green and M.F. Newman, Space groups and groups of prime power order I, Arch. Math.,35 (1980), 193–202.
L. Levai, On identities of A-analytic groups. Preprint, 1999.
L. Levai and L. Pyber, Coset-identities and dense free subgroups of profinite group. Preprint.
M.W. Liebeck and A. Shalev, The probability of generating a finite simple group, Geom. Ded.,56 (1995), 103–113.
V. Linchenko, Identities of Lie algebras with actions of Hopf algebras, Comm. Alg.,25 (1997), 3179–3187.
A. Lubotzky and A. Mann, Powerful p-groups. I, II, J. Alg., 105 (1987), 484–515.
A. Lubotzky and A. Mann, Groups of polynomial subgroup growth, Invent. Math., 104 (1991), 521–533.
A. Lubotzky, A. Mann and D. Segal, Finitely generated groups of polynomial subgroup growth, Israel. J. Math., 82 (1993), 363–371.
A. Lubotzky and A. Shalev, On some A-analytic pro-p Groupss, Israel J. Math., 85 (1994), 307–337.
W. Magnus, Über Gruppen und zugeordnete Liesche Ringe, Reine Angew. Math., 182 (1940), 142–159.
A. Mann, Positively finitely generated groups, Forum Math., 8 (1996), 429–459.
A. Mann, Finite groups containing many involutions, Proc. Amer. Math. Soc., 122 (1994), 383–385.
A. Mann and C. Martinez, The exponent of finite groups, Arch. Math., 67 (1996), 8–10.
A. Mann and C. Martinez, Groups nearly of prime exponent and nearly Engel Lie algebras, Arch. Math., 71 (1998), 5–11.
A. Mann and A. Shalev, Simple groups, maximal subgroups, and probabilistic aspects of profinite groups, Israel J. Math., 96 (1997) (Amitsur memorial issue), 449–468.
G.A. Margulis, Discrete Subgroups of Semisimple Lie groups, Springer, Berlin, 1991.
C. Martinez and E.I. Zelmanov, On pro-unipotent groups of finite width, to appear.
C. Martinez and E.I. Zelmanov, Nil algebras and unipotent groups of finite width, to appear.
O. Mathieu, Classification of simple graded Lie algebras of finite growth, Invent. Math.,108 (1990), 455–519.
Yu. Medvedev, Groups and Lie rings with almost regular automorphisms, J. Alg., 164 (1994), 877–885.
Yu. Medvedev, p-Groups, Lie p-rings and p-automorphisms, J. London Math. Soc., 58 (1998), 27–37.
Yu. Medvedev, p-Divided Lie rings and p-groups, J. London Math. Soc., 59 (1999), 787–798.
Yu. Medvedev, Compact Engel groups, Israel. J. Math., to appear.
C.F. Miller III and V.N. Obraztsov, Infinite periodic residually finite groups with all finite subgroups cyclic, in preparation.
J. Milnor, On fundamental groups of complete affinely flat manifolds, Adv. in Math., 25 (1977), 178–187.
S. Montgomery, Fixed Rings of Finite Automorphism Groups of Associative Rings, Lect. Notes Math. 818, Springer-Verlag, Berlin, 1980.
B.H. Neumann, Groups with automorphisms that leave only the identity element fixed, Arch. Math.,7 (1956), 1–5.
P.M. Neumann, Two combinatorial problems in group theory, Bull. London Math. Soc., 21 (1989), 456–458.
M.F. Newman and E.A. O’Brien, Classifying 2-groups by coclass, Trans. Amer. Math. Soc., 351 (1999), 131–169.
M.F. Newman, C. Schneider and A. Shalev, The entropy of graded algebras, J. Alg., 223 (2000), 85–100.
W.H. Patton, The Minimum Index for Subgroups in Some Classical Groups: A Generalization of a Theorem of Galois, Ph.D. Thesis, University of Illinois at Chicago Circle, Chicago, 1972.
V.M. Petrogradsky, On Lie algebras with non-integral q-dimensions, Proc. Amer. Math. Soc., 123 (1997), 649–656.
R. Pink, Compact subgroups of linear algebraic groups, J. Alg., 206 (1998), 438–504.
D.M. Riley and J.F. Semple, The coclass conjectures for restricted Lie algebras, Bull. London Math. Soc., 26 (1994), 431–437.
N.R. Rocco and P. Shumyatsky, On periodic groups having almost regular 2-elements, Proc. Edinburgh Math. Soc., 41 (1998), 385–391.
A. Rozhkov, The lower central series of one group of automorphisms of the trees, Mat. Zametki,60 (1996), 225–237.
C.M. Scoppola, Groups of prime power order as Frobenius—Wielandt complements, Trans. Amer. Math. Soc., 325 (1991), 855–874.
D. Segal, A footnote on residually finite groups, Isr. J. Math., 94 (1996), 1–5.
J.-P. Serre, Lie groups and Lie algebras (new edition), Lecture Notes in Math. 1500, Springer, Berlin, 1991.
A. Shalev, Growth functions, p-adic analytic groups, and groups of finite co-class, J. London Math. Soc., 46 (1992), 111–122.
A. Shalev, Combinatorial conditions in residually finite groups, II, J. Alg., 157 (1993), 51–62.
A. Shalev, On almost fixed-point-free automorphisms, J. Alg., 157 (1993), 271–282.
A. Shalev, Automorphisms of finite groups of bounded rank, Israel J. Math., 82 (1993), (Thompson issue), 395–404.
A. Shalev, The structure of finite p-groups: effective proof of the coclass conjectures, Invent. Math.,115 (1994), 315–345.
A. Shalev, Simple Lie algebras and Lie algebras of maximal class, Arch. Math., 63 (1994), 297–301.
A. Shalev, Finite p-groups, in Finite and Locally Finite Groups, eds: B. Hartley et al., NATO ASI Series, Kluwer, 1995, pp. 401–450.
A. Shalev, Groups whose subgroup growth is less than linear, International Journal of Algebra and Computation,7 (1997), 77–91.
A. Shalev, Centralizers in residually finite torsion groups, Proc. Amer. Math. Soc., 126 (1998), 3495–3499.
A. Shalev, Probabilistic group theory, Groups St Andrews 1997 in Bath, II, London Math. Soc. Lecture Note Series 261, Cambridge University Press, Cambridge, 1999, pp. 648–678.
A. Shalev, The orders of nonsingular derivations, J. Austral. Math. Soc., 67 (1999), 254–260.
A. Shalev and E.I. Zelmanov, Pro-p groups of finite coclass, Math. Proc. Cambr. Phil Soc., 111 (1992), 417–421.
A. Shalev and E.I. Zelmanov, Narrow Lie algebras: a coclass theory and a characterization of the Witt algebra, J. Alg., 189 (1997), 294–331.
P. Shumyatsky, On finite nilpotent groups having fixed point free automorphisms. Preprint, Brasilia, 1996.
P. Shumyatsky, Nilpotency of some Lie algebras associated with p-groups, Canad. J. Math., 51 (1999), 658–672.
P. Shumyatsky, On groups having a four-subgroup with finite centralizer, Quart. J. Math., 49 (1998), 491–499.
P. Shumyatsky, Centralizers in groups with finiteness conditions, J. Group Th., 1 (1998), 275–282.
P. Shumyatsky, On groups with commutators of bounded order, Proc. Amer. Math. Soc. 127 (1999), 2583–2586.
V.P. Shunkov, On periodic groups with an almost regular involution, Algebra and Logic, 11 (1972), 260–272.
L.W. Small, J.T. Stafford and R.B. Warfield, Affine algebras of Gelfand-Kirillov dimension one are PI, Math. Proc. Cambr. Phil. Soc., 97 (1985), 407–414.
J.G. Thompson, Finite groups with fixed-point-free automorphisms of prime order, Proc. Nat. Acad. Sci. U.S.A., 45 (1959), 578–581.
J.G. Thompson, Automorphisms of solvable groups, J. Alg., 1 (1964), 259–267.
M.R. Vaughan-Lee, The restricted Burnside problem, 2nd edition, Oxford Univ. Press, Oxford 1993.
M. Vergne, Réductibilité de la Variété des algèbres de Lie nilpotentes, C.R. Acad. Sc. Paris, 263 (1966), A4–A6.
M. Vergne, Cohomologie des algèbres de Lie nilpotentes. Application à l’étude de la variété des algèbres de Lie nilpotentes, C.R. Acad. Sc. Paris, 267 (1968), A867–A870.
J.S. Wilson, Two-generator conditions for residually finite groups, Bull. London Math. Soc., 23 (1991), 239–248.
J.S. Wilson and E.I. Zelmanov, Identities for Lie algebras of pro-p groups, J. Pure and Appl. Alg., 81 (1992), 103–109.
I.O. York, The Group of Formal Power Series under Substitution, Ph.D. Thesis, Nottingham, 1990.
E.I. Zelmanov, Engel Lie algebras, Dokl. Akad. Nauk SSSR, 292 (1987), 265–268.
E.I. Zelmanov, On some problems in the theory of groups and Lie algebras, Mat. Sb.,180 159–167. (in Russian)
E.I. Zelmanov, The solution of the restricted Burnside problem for groups of odd exponent, Math. USSR-Irv, 36 (1991), 41–60.
E.I. Zelmanov, The solution of the restricted Burnside problem for 2-groups, Mat. Sb., 182 (1991), 568–592.
E.I. Zelmanov, On periodic compact groups, Israel J. Math., 77 (1992), 83–95.
E.I. Zelmanov, Lie ring methods in the theory of nilpotent groups, in Groups ‘83, Galway-St Andrews, London Math. Soc. Lecture Note Series 212, 1995, 567–586.
A. Zubkov, Non-abelian free pro-p groups cannot be represented by 2-by-2 matrices, Sib. Math. J., 28 (1987), 742–747.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer Science+Business Media New York
About this chapter
Cite this chapter
Shalev, A. (2000). Lie Methods in the Theory of pro-p Groups. In: du Sautoy, M., Segal, D., Shalev, A. (eds) New Horizons in pro-p Groups. Progress in Mathematics, vol 184. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-1380-2_1
Download citation
DOI: https://doi.org/10.1007/978-1-4612-1380-2_1
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-7122-2
Online ISBN: 978-1-4612-1380-2
eBook Packages: Springer Book Archive