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LANDAU’S THEOREM, FIELDS OF VALUES FOR CHARACTERS, AND SOLVABLE GROUPS
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 104 / Issue 1 / February 2018
- Published online by Cambridge University Press:
- 23 December 2016, pp. 37-43
- Print publication:
- February 2018
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- Article
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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.
