On the number of conjugacy class sizes and character degrees in finite -groups

Authors:
Gustavo A. Fernández-Alcober and Alexander Moretó

Journal:
Proc. Amer. Math. Soc. **129** (2001), 3201-3204

MSC (1991):
Primary 20C15, 20D15

Published electronically:
March 29, 2001

MathSciNet review:
1844993

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Abstract | References | Similar Articles | Additional Information

In this note we prove that for any two integers there exist finite -groups of class such that and .

**1.**Bert Beisiegel,*Semi-extraspezielle 𝑝-Gruppen*, Math. Z.**156**(1977), no. 3, 247–254 (German). MR**0473004****2.**W. Burnside,*Theory of groups of finite order*, Dover Publications, Inc., New York, 1955. 2d ed. MR**0069818****3.**John Cossey and Trevor Hawkes,*Sets of 𝑝-powers as conjugacy class sizes*, Proc. Amer. Math. Soc.**128**(2000), no. 1, 49–51. MR**1641677**, 10.1090/S0002-9939-99-05138-2**4.**G.A. FERNÁNDEZ-ALCOBER, A. MORETÓ,*Groups with two extreme character degrees and their normal subgroups*, Trans. Amer. Math. Soc.**353**(2001), 2171-2192.**5.**Bertram Huppert,*Character theory of finite groups*, de Gruyter Expositions in Mathematics, vol. 25, Walter de Gruyter & Co., Berlin, 1998. MR**1645304****6.**I. M. Isaacs,*Sets of 𝑝-powers as irreducible character degrees*, Proc. Amer. Math. Soc.**96**(1986), no. 4, 551–552. MR**826479**, 10.1090/S0002-9939-1986-0826479-1**7.**I.M. ISAACS, ``Character Theory of Finite Groups", New York, Dover, 1994. CMP**94:14****8.**I. M. Isaacs and D. S. Passman,*A characterization of groups in terms of the degrees of their characters. II*, Pacific J. Math.**24**(1968), 467–510. MR**0241524**

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Additional Information

**Gustavo A. Fernández-Alcober**

Affiliation:
Departamento de Matemáticas, Universidad del País Vasco, 48080 Bilbao, Spain

Email:
mtpfealg@lg.ehu.es

**Alexander Moretó**

Affiliation:
Departamento de Matemáticas, Universidad del País Vasco, 48080 Bilbao, Spain

Email:
mtbmoqua@lg.ehu.es

DOI:
http://dx.doi.org/10.1090/S0002-9939-01-05946-9

Received by editor(s):
February 14, 2000

Received by editor(s) in revised form:
March 31, 2000

Published electronically:
March 29, 2001

Additional Notes:
Research of the second author was supported by a grant of the Basque Government and by the University of the Basque Country grant UPV 127.310-EB160/98.

Communicated by:
Stephen D. Smith

Article copyright:
© Copyright 2001
American Mathematical Society