Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-28T00:05:25.633Z Has data issue: false hasContentIssue false
  • English
  • Français

Lifts and vertex pairs in solvable groups

Part of: Representation theory of groups

Published online by Cambridge University Press:  04 January 2012

James P. Cossey  and
Mark L. Lewis
James P. Cossey
Affiliation:
Department of Theoretical and Applied Mathematics, University of Akron, Akron, OH 44325, USA (cossey@uakron.edu)
Mark L. Lewis
Affiliation:
Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA (lewis@math.kent.edu)
Rights & Permissions [Opens in a new window]

Abstract

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.

Suppose G is a p-solvable group, where p is odd. We explore the connection between lifts of Brauer characters of G and certain local objects in G, called vertex pairs. We show that if χ is a lift, then the vertex pairs of χ form a single conjugacy class. We use this to prove a sufficient condition for a given pair to be a vertex pair of a lift and to study the behaviour of lifts with respect to normal subgroups.

Keywords

liftsBrauer charactersfinite groupsrepresentationssolvable groups

MSC classification

Secondary: 20C20: Modular representations and characters 20C15: Ordinary representations and characters
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 1 , February 2012 , pp. 143 - 153
Copyright
Copyright © Edinburgh Mathematical Society 2011

References

1.Cossey, J. P., A construction of two distinct canonical sets of lifts of Brauer characters in a p-solvable group, Arch. Math. 87 (2006), 385–389.CrossRefGoogle Scholar
2.Cossey, J. P., Bounds on the number of lifts of a Brauer character in a p-solvable group, J. Alg. 312 (2007), 699–708.CrossRefGoogle Scholar
3.Cossey, J. P., Vertices of π-irreducible characters of groups of odd order, Commun. Alg. 36 (2008), 3972–3979.CrossRefGoogle Scholar
4.Cossey, J. P., Vertices and normal subgroups of solvable groups, J. Alg. 321 (2009), 2962–2969.CrossRefGoogle Scholar
5.Cossey, J. P., Navarro vertices and normal subgroups in groups of odd order, Rocky Mt. J. Math. (in press).Google Scholar
6.Cossey, J. P., Lewis, M. L., and Navarro, G., The number of lifts of a Brauer character with normal vertex, J. Alg. 328 (2011), 484–487.CrossRefGoogle Scholar
7.Green, J. A., On the indecomposable representations of a finite group, Math. Z. 70 (1959), 430–445.CrossRefGoogle Scholar
8.Huppert, B., Character theory of finite groups (Walter de Gruyter, Berlin, 1998).CrossRefGoogle Scholar
9.Isaacs, I. M., Character theory of finite groups (Academic Press, San Diego, 1976).Google Scholar
10.Isaacs, I. M., Characters of π-separable groups, J. Alg. 86 (1984), 98–128.CrossRefGoogle Scholar
11.Isaacs, I. M., Induction and restriction of π-special characters, Can. J. Math. 37 (1986), 576–604.CrossRefGoogle Scholar
12.Isaacs, I. M., Fong characters in π-separable groups, J. Alg. 99 (1986), 89–107.CrossRefGoogle Scholar
13.Isaacs, I. M., Partial characters of π-separable groups, Progr. Math. 95 (1991), 273–287.Google Scholar
14.Isaacs, I. M., Characters and Hall subgroups of groups of odd order, J. Alg. 157 (1993), 548–561.CrossRefGoogle Scholar
15.Isaacs, I. M., Characters and sets of primes for solvable groups, in Finite and locally finite groups, NATO Advanced Science Institute Series C: Mathematical and Physical Sciences, Volume 471, pp. 347–376 (Kluwer, Dordrecht, 1995).CrossRefGoogle Scholar
16.Isaacs, I. M. and Navarro, G., Weights and vertices for characters of π-separable groups, J. Alg. 177 (1995), 339–366.CrossRefGoogle Scholar
17.Laradji, A., Vertices of simple modules and normal subgroups of p-solvable groups, Arch. Math. 79 (2002), 418–422.CrossRefGoogle Scholar
18.Manz, O. and Wolf, T. R., Representation of solvable groups (Cambridge University Press, 1993).CrossRefGoogle Scholar
19.Navarro, G., Vertices for characters of p-solvable groups, Trans. Am. Math. Soc. 354 (2002), 2759–2773.CrossRefGoogle Scholar
20.Navarro, G., Actions and characters in blocks, J. Alg. 275 (2004), 471–480.CrossRefGoogle Scholar
21.Navarro, G., Modularly irreducible characters and normal subgroups, Osaka J. Math. 48(2) (2011), 329–332.Google Scholar