Skip to main content Accessibility help
×
Hostname: page-component-788cddb947-pt5lt Total loading time: 0 Render date: 2024-10-08T10:27:22.599Z Has data issue: false hasContentIssue false

Structure of finite groups having few conjugacy class sizes

Published online by Cambridge University Press:  05 July 2011

Antonio Beltrán
Affiliation:
Universitat Jaume I, Spain
María José Felipe
Affiliation:
Universidad Politécnica de Valencia, Spain
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
M. R. Quick
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
C. M. Roney-Dougal
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
G. Traustason
Affiliation:
University of Bath
Get access

Summary

Abstract

The structure of finite groups has a significant influence on the conjugacy class sizes and reciprocally is also influenced by them. In this paper we present some classic and recent contributions which have been obtained during the last forty years related to the structure and properties of those groups having few conjugacy class sizes.

Introduction

The structure of a finite group strongly controls and at the same time is controlled by the sizes of its conjugacy classes and this relation also occurs for the degrees of its irreducible characters. If G is a finite group, we will denote by cs(G) the set of the conjugacy class sizes of G and by cd(G) the set of its irreducible character degrees. It is accepted that there exists certain parallelism between the results accomplished on the group structure from its character degrees and the ones obtained from its class sizes, although the techniques employed to show them may be completely different. We focus our attention to the cardinals of these sets. The fact that they are small for a group may provide a lot of information about it, but we observe that apparently there seems to be no relation between them. For instance, we have that G = SL(2, 5) satisfies |cs(G)| = 4 and |cd(G)| = 6, and that G = A7 satisfies |cs(G)| = 8 and |cd(G)| = 7, whereas the equality holds for G = A5 since |cs(G)| = |cd(G)| = 4.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] E., Alemany, A., Beltrán, M.J., Felipe, Finite groups with two p-regular conjugacy class lengths II, Bull. Austral. Math. Soc. 79 (2009), 419–425.Google Scholar
[2] B., Amberg, L., Kazarin, On the product of a nilpotent group and a group with nontrivial center, J. Algebra 311 (2007), 69–95.Google Scholar
[3] M., Aschbacher, Finite Group Theory, Cambridge University Press, New York, 1986.Google Scholar
[4] A., Beltrán, M.J., Felipe, Finite groups with two p-regular conjugacy class lengths, Bull. Austral. Math. Soc. 67 (2003), 163–169.Google Scholar
[5] A., Beltrán, M.J., Felipe, Variations on a theorem by Alan Camina on conjugacy class sizes, J. Algebra 296 (2006), 253–266.Google Scholar
[6] A., Beltrán, M.J., Felipe, Some class size conditions implying solvability of finite groups, J. Group Theory 9 (2006), 787–797.Google Scholar
[7] A., Beltrán, M.J., Felipe, Structure of finite groups under certain arithmetical conditions on class sizes, J. Algebra 319 (2008), 897–910.Google Scholar
[8] A., Beltrán, M.J., Felipe, The structure of groups with three class sizes, J. Group Theory, to appear.
[9] A., Beltrán, M.J., Felipe, Finite groups with four conjugacy class sizes. Submitted.
[10] A., Beltrán, M.J., Felipe, On the solvability of finite groups with four class sizes. Submitted.
[11] E.A., Bertram, M., Herzog, A., Mann, On a graph related to conjugacy classes of groups, Bull. London Math. Soc. 22 (1990), 569–575.Google Scholar
[12] A.R., Camina, Arithmetical conditions on the conjugacy class numbers of a finite group, J. London Math. Soc. 2(5) (1972), 127–132.Google Scholar
[13] A.R., Camina, Finite groups of conjugate rank 2, Nagoya Math. J. 53 (1974), 47–57.Google Scholar
[14] A.R., Camina, R.D., Camina. Coprime conjugacy class sizes, Asian-Eur. J. Math. 2 (2009), 183–190.Google Scholar
[15] S., Dolfi, E., Jabara, The structure of finite groups of conjugate rank 2, Bull. London Math. Soc., to appear.
[16] E., Fisman, Z., Arad, A proof of Szep's conjecture on non-simplicity of certain finite groups, J. Algebra 108 (1987), 340–354.Google Scholar
[17] D., Gorenstein, J.H., Walter, On finite groups with dihedral Sylow 2-subgroups, Illinois J. Math. 6 (1962), 553–593.Google Scholar
[18] B., Huppert, Character Theory of Finite Groups, De Gruyter Expositions in Mathematics 25, Walter de Gruyter & Co, Berlin, 1998.Google Scholar
[19] K., Ishikawa, On finite p-groups which have only two conjugacy lengths, Israel J. Math. 129 (2002), 119–123.Google Scholar
[20] I.M., Isaacs, Character Theory of Finite Groups, Dover, New York, 1994.Google Scholar
[21] I.M., Isaacs, Groups with many equal classes, Duke Math. J. 37 (1970), 501–506.Google Scholar
[22] I.M., Isaacs, Subgroups generated by small classes in finite groups, Proc. Amer. Math. Soc. 136 (2008), no. 7, 2299–2301.Google Scholar
[23] N., Itô, On finite groups with given conjugate types I, Nagoya Math. 6 (1953), 17–28.Google Scholar
[24] N., Itô, On finite groups with given conjugate types II, Osaka J. Math. 7 (1970), 231–251.Google Scholar
[25] N., Itô, On finite groups with given conjugate types III, Math. Z. 117 (1970), 267–271.Google Scholar
[26] N., Itô, Simple groups of conjugate type rank 4, J. Algebra 20 (1972), 226–249.Google Scholar
[27] N., Itô, Simple groups of conjugate type rank 5, J. Math. Kyoto Univ. 13 (1973), 171–190.Google Scholar
[28] L., Kazarin, On groups with isolated conjugacy classes, Izvestiya Vysshikh Uchebnyckh Zavedeniy Matematika no. 7 (1981), 40–45.Google Scholar
[29] L.S., Kazarin, On the product of an abelian group and a group with nontrivial center, Manuscript No.3565-81, deposited at VINITI, 1981. (Russian)
[30] H.G., Knoche, Über den Frobeniusschen Klassbegriff in nilpotenten Gruppen, Math. Z. 55 (1951), 71–83.Google Scholar
[31] M.L., Lewis, Derived lengths of solvable groups having five irreducible character degrees II, Algeb. Represent. Theory 5 (2002), 277–304.Google Scholar
[32] G., Malle, A., Moretó, Nonsolvable groups with few character degrees, J. Algebra 294 (2005), 117–126.Google Scholar
[33] A., Mann, Conjugacy class sizes in finite groups, J. Austral. Math. Soc. 85 (2008), 251–255.Google Scholar
[34] A., Mann, Elements of minimal breadth in finite p-groups and Lie algebras, J. Austral. Math. Soc. 81 (2006), 209–214.Google Scholar
[35] A., Mann, Groups with few class sizes and centraliser equality subgroup, Israel J. Math. 142 (2004), 367–380.Google Scholar
[36] G., Qian, W., Shi, A note on character degrees of finite groups, J. Group Theory 7 (2004), 187–196.Google Scholar
[37] J., Rebmann, F-Gruppen, Arch. Math. (Basel) 22 (1971), 225–230.Google Scholar
[38] J.M., Riedl, Fitting heights of odd order groups with few character degrees, J. Algebra 267 (2003), 421–442.Google Scholar

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×