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On groups having a $ p$-constant character

Authors: Silvio Dolfi, Emanuele Pacifici and Lucía Sanus
Journal: Proc. Amer. Math. Soc. 149 (2021), 107-120
MSC (2010): Primary 20C15
Published electronically: October 20, 2020
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Abstract: Let $ G$ be a finite group, and $ p$ a prime number; a character of $ G$ is called $ p$-constant if it takes a constant value on all the elements of $ G$ whose order is divisible by $ p$. This is a generalization of the very important concept of characters of $ p$-defect zero. In this paper, we characterize the finite $ p$-solvable groups having a faithful irreducible character that is $ p$-constant and not of $ p$-defect zero, and we will show that a non-$ p$-solvable group with this property is an almost-simple group.

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Silvio Dolfi
Affiliation: Dipartimento di Matematica e Informatica U. Dini, Università degli Studi di Firenze, viale Morgagni 67/a, 50134 Firenze, Italy

Emanuele Pacifici
Affiliation: Dipartimento di Matematica F. Enriques, Università degli Studi di Milano, via Saldini 50, 20133 Milano, Italy

Lucía Sanus
Affiliation: Departament de Matemàtiques, Facultat de Matemàtiques, Universitat de València, 46100 Burjassot, València, Spain

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