Advertisement

Analyticity and Growth of Pro-p-Groups

  • A. Caranti
Chapter
  • 141 Downloads
Part of the NATO ASI Series book series (ASIC, volume 333)

Abstract

Our subject are the algebraic properties of p-adic analytic groups. We present first the theory of finite and pro-finite powerful p-groups, as developed by Lubotzky and Mann. Then we show, following work by Lubotzky, Mann and Segal, how this theory, and previous results of Lubotzky characterizing linear groups, can be applied to the study of residually finite groups of finite (Prüfer) rank. Finally, we present some recent work of Lubotzky, Mann, Segal and Du Sautoy related to polynomial subgroup growth.

Keywords

Normal Subgroup Sylow Subgroup Finite Index Finite Rank Open Subgroup 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    F. J. Grunewald, D. Segal and G. C. Smith, Subgroups of finite index in nilpotent groups. Invent. Math. 93 (1988), 185–223.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    M. Hall, The Theory of Groups. Chelsea, New York, 1976.zbMATHGoogle Scholar
  3. [3]
    B. Huppert, Endliche Gruppen I. Springer, Berlin, 1967.zbMATHCrossRefGoogle Scholar
  4. [4]
    I. Ilani, Counting finite index subgroups and the P. Hall enumeration principle. Isr. J. Math. 68 (1989), 18–26.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5] W. M. Kantor and A. Lubotzky, The probability of generating a finite classical group, to appear in Geom. Ded.Google Scholar
  6. [6]
    M. Lazard, Groupes analytiques p-adiques. Publ. Math. I.H.E.S. 26 (1965), 389–603.MathSciNetzbMATHGoogle Scholar
  7. [7]
    A. Lubotzky, A group theoretic characterization of linear groups. J. Algebra 113 (1988), 207–214.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    A. Lubotzky and A. Mann, Powerful p-groups. I. Finite groups. J. Algebra 105 (1987), 484–505.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    A. Lubotzky and A. Mann, Powerful p-groups. II. p-adic analytic groups. J. Algebra 105 (1987), 506–515.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    A. Lubotzky and A. Mann, Residually finite groups of finite rank. Math. Proc. Cam. Phil. Soc. 106 (1989), 385–388.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    A. Lubotzky and A. Mann, On groups of polynomial subgroup growth, to appear in Invent. math.Google Scholar
  12. [12]
    A. Mann and D. Segal, Uniform finiteness conditions in residually finite groups, to appear in Proc. Camb. Phil. Soc.Google Scholar
  13. [13]
    D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups. (Part 1 and 2) Springer, Berlin, 1972.Google Scholar
  14. [14]
    D. Segal, Polycylic groups. Cambridge University Press, Cambridge, 1983.CrossRefGoogle Scholar
  15. [15]
    D. Segal, Subgroups of finite index in soluble groups. In: Groups, St. Andrews 1985, Ed. C. M. Campbell, E. E. Robertson, pp. 307–319, Cambridge University Press, Cambridge, 1986.Google Scholar
  16. [16]
    J.-P. Serre, A Course in Arithmetic. Springer, New York, 1973.zbMATHCrossRefGoogle Scholar
  17. [17]
    S. S. Shatz, Profinite Groups, Arithmetic, and Geometry. Princeton, 1972.Google Scholar
  18. [18]
    S. J. Tobin, Groups with exponent four. In: Groups, St. Andrews 1981, Ed. C. M. Campbell, E. F. Robertson, pp. 81–136, Cambridge University Press, Cambridge, 1982.CrossRefGoogle Scholar
  19. [19]
    B. A. F. Wehrfritz, Infinite Linear Groups. Springer, Berlin, 1967.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1991

Authors and Affiliations

  • A. Caranti
    • 1
  1. 1.Dipartimento di MatematicaUniversità degli Studi di TrentoPovo (Trento)Italy

Personalised recommendations