Kernels of minimal characters of solvable groups (Q6657212)

Let \(G\) be a finite solvable group and \(\mathrm{Irr}(G)\) be the set of irreducible complex characters of \(G\). A classical theorem of \textit{D. M. Broline} [J. Algebra 45, 83--87 (1977; Zbl 0367.20003)] and \textit{S. C. Garrison} [J. Algebra 32, 623--628 (1974; Zbl 0294.20007)] implies that if \(\chi \in \mathrm{Irr}(G)\) has maximal degree, then \(\ker \chi\) is nilpotent.\N\NThe goal of the paper under review is to consider irreducible characters at the other extreme. Let \(m(G)=\min \{ \chi(1) \mid \chi \in \mathrm{Irr}(G), \, \chi(1)>1 \}\); a character \(\chi \in \mathrm{Irr}(G)\) is minimal if \(\chi(1)=m(G)\). The main result of the author is Theorem A: Let \(G\) be a solvable finite group such that \(m(G)\) is odd. If \(\chi \in \mathrm{Irr}(G)\) is a minimal character, then \(G/ \ker\chi\) is nilpotent-by-abelian.