2-length and rational characters of odd degree.
Let \(G\) be a finite solvable group of 2-length \(l\). \textit{G. Navarro} and \textit{P. H. Tiep} [Trans. Am. Math. Soc. 360, No. 5, 2443-2465 (2008; Zbl 1137.20009)] have shown that \(l\) is less than the number of rational irreducible characters of odd degree. In the paper under review the author improves this linear upper bound to a logarithmic bound by proving that the number of rational irreducible characters of \(G\) of odd degree is at least \(2^l\). He also provides examples showing that for every \(l\) this bound is best possible. Moreover, the main result together with the above-mentioned work of Navarro and Tiep easily implies the following group theoretic consequence: If \(P\) is a Sylow 2-subgroup of \(G\), then the number of \(N_G(P)\)-orbits on \(P/\Phi(P)\) is at least \(2^l\). -- In this reviewer's opinion, it would be interesting to (1) find a purely group theoretical proof for this fact and (2) to extend it to odd primes.
numbers of rational irreducible characters
scientific article; zbMATH DE number 5886764
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