Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Character degrees and nilpotence class of finite $p$-groups: An approach via pro-$p$ groups


Authors: A. Jaikin-Zapirain and Alexander Moretó
Journal: Trans. Amer. Math. Soc. 354 (2002), 3907-3925
MSC (2000): Primary 20C15; Secondary 20E18
DOI: https://doi.org/10.1090/S0002-9947-02-02992-6
Published electronically: April 12, 2002
MathSciNet review: 1926859
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $\mathcal{S}$ be a finite set of powers of $p$ containing 1. It is known that for some choices of $\mathcal{S}$, if $P$ is a finite $p$-group whose set of character degrees is $\mathcal{S}$, then the nilpotence class of $P$ is bounded by some integer that depends on $\mathcal{S}$, while for some other choices of $\mathcal{S}$ such an integer does not exist. The sets of the first type are called class bounding sets. The problem of determining the class bounding sets has been studied in several papers whose results made it tempting to conjecture that a set $\mathcal{S}$ is class bounding if and only if $p\notin\mathcal{S}$. In this article we provide a new approach to this problem. Our main result shows the relevance of certain $p$-adic space groups in this problem. With its help, we are able to prove some results that provide new class bounding sets. We also show that there exist non-class-bounding sets $\mathcal{S}$ such that $p\notin\mathcal{S}$.


References [Enhancements On Off] (What's this?)

  • 1. J. D. DIXON, M. P. F. DU SAUTOY, A. MANN, D. SEGAL, ``Analytic Pro-$p$ Groups", Second edition. Cambridge Studies in Advanced Mathematics, 61. Cambridge University Press, Cambridge, 1999. MR 2000m:20039
  • 2. R. M. GURALNICK, On the number of generators of a finite group, Arch. Math. 53 (1989), 521-523. MR 90m:20027
  • 3. B. HUPPERT, ``Endliche Gruppen", Springer-Verlag, Berlin-New York, 1967. MR 37:302
  • 4. B. HUPPERT, A remark on the character-degrees of some $p$-groups, Arch. Math. 59 (1992), 313-318. MR 93g:20016
  • 5. I. M. ISAACS, Sets of $p$-powers as irreducible character degrees, Proc. Amer. Math. Soc. 96 (1986), 551-552. MR 87d:20013
  • 6. I. M. ISAACS, ``Character Theory of Finite Groups", Dover, New York, 1994. MR 57:417 (original ed.)
  • 7. I. M. ISAACS, Characters of groups associated with finite algebras, J. Algebra 177 (1995), 708-730. MR 96k:20011
  • 8. I. M. ISAACS, G. KNUTSON, Irreducible character degrees and normal subgroups, J. Algebra 199 (1998), 302-326. MR 98m:20013
  • 9. I. M. ISAACS, A. MORETÓ, The character degrees and nilpotence class of a $p$-group, J. Algebra 238 (2001), 827-842. MR 2002a:20008
  • 10. I. M. ISAACS, D. S. PASSMAN, A characterization of groups in terms of the degrees of their characters II, Pacific J. Math. 24 (1968), 467-510. MR 39:2864
  • 11. I. M. ISAACS, M. C. SLATTERY, Character degree sets that do not bound the class of a $p$-group, to appear in Proc. Amer. Math. Soc.
  • 12. T. M. KELLER, Orbit sizes and character degrees, Pacific J. Math. 187 (1999), 317-332. MR 99m:20013
  • 13. G. KLAAS, C. R. LEEDHAM-GREEN, W. PLESKEN, ``Linear Pro-$p$Groups of Finite Width", Lecture Notes in Mathematics 1674, Springer-Verlag, Berlin, 1997. MR 98m:20028
  • 14. L. KOVACS, On finite soluble groups, Math. Z. 103 (1968), 37-39. MR 36:6506
  • 15. C. R. LEEDHAM-GREEN, S. MCKAY, 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. (3) 52 (1986), 73-94. MR 87g:20036
  • 16. C. R. LEEDHAM-GREEN, M. F. NEWMAN, Space groups and groups of prime-power order. I., Arch. Math. 35 (1980), 193-202. MR 81m:20029
  • 17. A. LUCCHINI, A bound on the number of generators of a finite group, Arch. Math. 53 (1989), 313-317. MR 90m:20026
  • 18. A. MANN, Generators of $2$-groups, Israel J. Math. 10 (1971), 158-159. MR 45:5221
  • 19. A. MANN, Minimal characters of $p$-groups, J. Group Theory 2 (1999), 225-250. MR 2000f:20007
  • 20. A. PREVITALI, Orbit lengths and character degrees in $p$-Sylow subgroups of some classical Lie groups, J. Algebra 177 (1995), 658-675. MR 96m:20013
  • 21. J. M. RIEDL, Fitting heights of solvable groups with few character degrees, J. Algebra 233 (2000), 287-308. CMP 1 793 598
  • 22. M. C. SLATTERY, Character degrees and nilpotence class in $p$-groups, J. Austral Math. Soc. (Series A) 57 (1994), 76-80. MR 95d:20013
  • 23. J. S. WILSON, ``Profinite Groups", London Math. Soc. Monographs, New Series, 19, The Clarendon Press, Oxford University Press, 1998. MR 2000j:20048

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 20C15, 20E18

Retrieve articles in all journals with MSC (2000): 20C15, 20E18


Additional Information

A. Jaikin-Zapirain
Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Cantoblanco Ciudad Universitaria, 28049 Madrid, Spain
Email: ajaikin@uam.es

Alexander Moretó
Affiliation: Departamento de Matemáticas, Universidad del País Vasco, Apartado 644, 48080 Bilbao, Spain
Email: mtbmoqua@lg.ehu.es

DOI: https://doi.org/10.1090/S0002-9947-02-02992-6
Received by editor(s): July 18, 2001
Received by editor(s) in revised form: December 17, 2001
Published electronically: April 12, 2002
Additional Notes: Research of the first author partially supported by DGICYT. Research of the second author supported by the Basque Government and the University of the Basque Country.
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society