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A NOTE ON DERIVED LENGTH AND CHARACTER DEGREES

Part of: Representation theory of groups

Published online by Cambridge University Press:  13 July 2020

BURCU ÇINARCI Open the ORCID record for BURCU ÇINARCI [Opens in a new window]  and
TEMHA ERKOÇ Open the ORCID record for TEMHA ERKOÇ [Opens in a new window]
BURCU ÇINARCI
Affiliation:
Maritime Faculty, Piri Reis University, 34940 Istanbul, Turkey email burcu-cinarci@hotmail.com
TEMHA ERKOÇ*
Affiliation:
Department of Mathematics, Faculty of Science,Istanbul University, 34134 Istanbul, Turkey email erkoctemha@gmail.com
*
✉erkoctemha@gmail.com
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Abstract

Isaacs and Seitz conjectured that the derived length of a finite solvable group $G$ is bounded by the cardinality of the set of all irreducible character degrees of $G$. We prove that the conjecture holds for $G$ if the degrees of nonlinear monolithic characters of $G$ having the same kernels are distinct. Also, we show that the conjecture is true when $G$ has at most three nonlinear monolithic characters. We give some sufficient conditions for the inequality related to monolithic characters or real-valued irreducible characters of $G$ when the commutator subgroup of $G$ is supersolvable.

Keywords

solvable groupsderived lengthreal and monolithic charactersTaketa inequalityIsaacs–Seitz conjecture

MSC classification

Primary: 20C15: Ordinary representations and characters
Secondary: 20C20: Modular representations and characters
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 103 , Issue 2 , April 2021 , pp. 271 - 277
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

The work of the authors was supported by the Scientific Research Projects Coordination Unit of Istanbul University (project number 27148).

References

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