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LANDAU’S THEOREM, FIELDS OF VALUES FOR CHARACTERS, AND SOLVABLE GROUPS

Part of: Representation theory of groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  23 December 2016

MARK L. LEWIS Open the ORCID record for MARK L. LEWIS [Opens in a new window]
MARK L. LEWIS*
Affiliation:
Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA email lewis@math.kent.edu
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Abstract

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When $G$ is a finite solvable group, we prove that $|G|$ can be bounded by a function in the number of irreducible characters with values in fields where $\mathbb{Q}$ is extended by prime power roots of unity. This gives a character theory analog for solvable groups of a theorem of Héthelyi and Külshammer that bounds the order of a finite group in terms of the number of conjugacy classes of elements of prime power order. In particular, we obtain for solvable groups a generalization of Landau’s theorem.

Keywords

Landau’s theoremsolvable groupsirreducible charactersfields of value

MSC classification

Primary: 20C15: Ordinary representations and characters
Secondary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks 20E45: Conjugacy classes
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 104 , Issue 1 , February 2018 , pp. 37 - 43
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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