Hostname: page-component-7b7567c8c-kwt9d Total loading time: 0 Render date: 2026-11-14T16:53:45.458Z Has data issue: false hasContentIssue false
  • English
  • Français

Character degrees and derived length in p-groups

Published online by Cambridge University Press:  18 May 2009

Michael C. Slattery
Michael C. Slattery
Affiliation:
Department of Mathematics, Statistics and Computer Science, Milwaukee, WI 53233
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

There are a number of theorems which bound d.l.(G), the derived length of a group G, in terms of the size of the set c.d.(G) of irreducible character degrees of G assuming that G is in some particular class of solvable groups ([1], [3], [4], [7]). For instance, Gluck [4] shows that d.l.(G)≤2 |c.d.(G)| for any solvable group, whereas Berger [1] shows that d.l.(G)≤|c.d.(G)| if G has odd order. One of the oldest (and smallest) such bounds is a theorem of Taketa [7] which says that d.l.(G)≤|c.d.(G)| if G is an M-group. Most of the existing theorems are an attempt to extend Taketa's bound to all solvable groups. However, it is not even known for M-groups whether or not this is the best possible bound. This suggests that given a class of solvable groups one might try to find the maximum derived length of a group with n character degrees (i.e. the best possible bound).

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 30 , Issue 2 , May 1988, pp. 221 - 230
Copyright
Copyright © Glasgow Mathematical Journal Trust 1988

References

1.Berger, T. R., Character degrees and derived length in groups of odd order, J. Algebra 39 (1976), 199–207.CrossRefGoogle Scholar
2.Cannon, J. J., An introduction to the group theory language CAYLEY, Computational Group Theory (Durham 1982), Atkinson, M. (ed.) (Academic Press, (1984)).Google Scholar
3.Garrison, S., On groups with a small number of character degrees, Ph.D. Thesis, University of Wisconsin (1973).Google Scholar
4.Gluck, D., Bounding the number of character degrees of a solvable group, J. London Math. Soc. (2) 31 (1985), 457–462.CrossRefGoogle Scholar
5.James, R., The groups of order p 6 (pan odd prime), Math. Comp. 34 (1980), 613–637.Google Scholar
6.Slattery, M., Character degrees of p-groups: a case study. The CAYLEY bulletin, University of Sydney, to appear.Google Scholar
7.Taketa, K., Uber die Gruppen deren Darstellungen sich sämtlich auf monomiale, Proc. Acad. Tokyo 6 (1930), 31–33.Google Scholar