On groups having a -constant character
Authors:
Silvio Dolfi, Emanuele Pacifici and Lucía Sanus
Journal:
Proc. Amer. Math. Soc. 149 (2021), 107-120
MSC (2010):
Primary 20C15
DOI:
https://doi.org/10.1090/proc/15256
Published electronically:
October 20, 2020
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: Let be a finite group, and
a prime number; a character of
is called
-constant if it takes a constant value on all the elements of
whose order is divisible by
. This is a generalization of the very important concept of characters of
-defect zero. In this paper, we characterize the finite
-solvable groups having a faithful irreducible character that is
-constant and not of
-defect zero, and we will show that a non-
-solvable group with this property is an almost-simple group.
- [1] Bertram Huppert, Zweifach transitive, auflösbare Permutationsgruppen, Math. Z. 68 (1957), 126–150 (German). MR 94386, https://doi.org/10.1007/BF01160336
- [2] Bertram Huppert and Norman Blackburn, Finite groups. III, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 243, Springer-Verlag, Berlin-New York, 1982. MR 662826
- [3] I. Martin Isaacs, Character theory of finite groups, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Pure and Applied Mathematics, No. 69. MR 0460423
- [4] Olaf Manz and Thomas R. Wolf, Representations of solvable groups, London Mathematical Society Lecture Note Series, vol. 185, Cambridge University Press, Cambridge, 1993. MR 1261638
- [5] Gabriel Navarro, Character theory and the McKay conjecture, Cambridge Studies in Advanced Mathematics, vol. 175, Cambridge University Press, Cambridge, 2018. MR 3753712
- [6] G. Navarro, Characters and blocks of finite groups, London Mathematical Society Lecture Note Series, vol. 250, Cambridge University Press, Cambridge, 1998. MR 1632299
- [7] Gabriel Navarro and Geoffrey R. Robinson, Irreducible characters taking root of unity values on 𝑝-singular elements, Proc. Amer. Math. Soc. 140 (2012), no. 11, 3785–3792. MR 2944719, https://doi.org/10.1090/S0002-9939-2012-11242-0
- [8] D. S. Passman, 𝑝-solvable doubly transitive permutation groups, Pacific J. Math. 26 (1968), 555–577. MR 236252
- [9] Marco Antonio Pellegrini, Irreducible 𝑝-constant characters of finite reflection groups, J. Group Theory 20 (2017), no. 5, 911–923. MR 3692055, https://doi.org/10.1515/jgth-2016-0059
- [10] Marco A. Pellegrini and Alexandre Zalesski, Irreducible characters of finite simple groups constant at the 𝑝-singular elements, Rend. Semin. Mat. Univ. Padova 136 (2016), 35–50. MR 3593541, https://doi.org/10.4171/RSMUP/136-4
- [11] Michio Suzuki, Group theory. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 247, Springer-Verlag, Berlin-New York, 1982. Translated from the Japanese by the author. MR 648772
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Additional Information
Silvio Dolfi
Affiliation:
Dipartimento di Matematica e Informatica U. Dini, Università degli Studi di Firenze, viale Morgagni 67/a, 50134 Firenze, Italy
Email:
silvio.dolfi@unifi.it
Emanuele Pacifici
Affiliation:
Dipartimento di Matematica F. Enriques, Università degli Studi di Milano, via Saldini 50, 20133 Milano, Italy
Email:
emanuele.pacifici@unimi.it
Lucía Sanus
Affiliation:
Departament de Matemàtiques, Facultat de Matemàtiques, Universitat de València, 46100 Burjassot, València, Spain
Email:
lucia.sanus@uv.es
DOI:
https://doi.org/10.1090/proc/15256
Received by editor(s):
May 16, 2020
Received by editor(s) in revised form:
June 5, 2020
Published electronically:
October 20, 2020
Additional Notes:
The first two authors were partially supported by the Italian INdAM-GNSAGA
The research of the third author was partially supported by Ministerio de Ciencia e Innovación PID2019-103854GB-I00 and FEDER funds.
The third author thanks Università degli Studi di Firenze for the financial support.
Dedicated:
Dedicated to Carlo Casolo
Communicated by:
Martin Liebeck
Article copyright:
© Copyright 2020
American Mathematical Society