On the number of irreducible real-valued characters of a finite group

We prove that there exists an integer-valued function f on positive integers such that if a finite group G has at most k real-valued irreducible characters, then |G/Sol(G)|≤f(k), where Sol(G) denotes the largest solvable normal subgroup of G. In the case k=5, we further classify G/Sol(G). This partl...

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Bibliographic Details
Published in:Journal of algebra 2020-08, Vol.555, p.275-288
Main Authors: Hung, Nguyen Ngoc, Schaeffer Fry, A.A., Tong-Viet, Hung P., Vinroot, C. Ryan
Format: Article
Language:English
Subjects:
Finite groups
Real elements
Real-valued characters
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Summary:We prove that there exists an integer-valued function f on positive integers such that if a finite group G has at most k real-valued irreducible characters, then |G/Sol(G)|≤f(k), where Sol(G) denotes the largest solvable normal subgroup of G. In the case k=5, we further classify G/Sol(G). This partly answers a question of Iwasaki [15] on the relationship between the structure of a finite group and its number of real-valued irreducible characters.
ISSN:0021-8693
1090-266X
DOI:10.1016/j.jalgebra.2020.03.008