Archiv der Mathematik

, 96:321 | Cite as

On the rank of compact p-adic Lie groups



The rank of a profinite group G is the basic invariant \({{\rm rk}(G):={\rm sup}\{d(H) \mid H \leq G\}}\), where H ranges over all closed subgroups of G and d(H) denotes the minimal cardinality of a topological generating set for H. A compact topological group G admits the structure of a p-adic Lie group if and only if it contains an open pro-p subgroup of finite rank. For every compact p-adic Lie group G one has rk(G) ≥ dim(G), where dim(G) denotes the dimension of G as a p-adic manifold. In this paper we consider the converse problem, bounding rk(G) in terms of dim(G). Every profinite group G of finite rank admits a maximal finite normal subgroup, its periodic radical π(G). One of our main results is the following. Let G be a compact p-adic Lie group such that π(G) = 1, and suppose that p is odd. If \(\{g \in G \mid g^{p-1}=1 \}\) is equal to {1}, then rk(G) = dim(G).

Mathematics Subject Classification (2000)

Primary 20E18 Secondary 22E20 


Compact p-adic Lie group Rank Dimension Number of generators 


  1. 1.
    J. D. Dixon et al., Analytic pro-p groups, 2nd ed., Cambridge Studies in Advanced Mathematics 61, Cambridge University Press, Cambridge, 1999.Google Scholar
  2. 2.
    Dixon J.D., Kovács L.G.: Generating finite nilpotent irreducible linear groups. Quart. J. Math. Oxford 44, 1–15 (1993)CrossRefMATHGoogle Scholar
  3. 3.
    González-Sánchez J., Klopsch B.: On w-maximal groups. J. Algebra 328, 155–166 (2011)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Guralnick R.: On the number of generators of a finite group. Arch. Math. (Basel) 53, 521–523 (1989)MATHMathSciNetGoogle Scholar
  5. 5.
    Heller A., Reiner I.: Representations of cyclic groups in rings of integers I. Ann. of Math. 76, 73–92 (1962)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Isaacs I.M.: The number of generators of a linear p-group. Can. J. Math. 24, 851–858 (1972)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Kovács L.G., Robinson G.R.: Generating finite completely reducible linear groups. Proc. Amer. Math. Soc. 112, 357–364 (1991)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Laffey T.J.: The minimum number of generators of a finite p-group. Bull. London Math. Soc. 5, 288–290 (1973)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Leedham-Green C.R., Plesken W.: Some remarks on Sylow subgroups of general linear groups. Math. Z. 191, 529–535 (1986)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Lucchini A.: A bound on the number of generators of a finite group. Arch. Math. (Basel) 53, 313–317 (1989)MATHMathSciNetGoogle Scholar
  11. 11.
    Lucchini A.: Some questions on the number of generators of a finite group. Rend. Sem. Mat. Univ. Padova 83, 202–222 (1990)MathSciNetGoogle Scholar
  12. 12.
    Lucchini A., Menegazzo F., Morigi M.: On the number of generators and composition length of finite linear groups. J. Algebra 243, 427–447 (2001)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of MathematicsRoyal Holloway, University of LondonEghamUK

Personalised recommendations