The largest character degree and the Sylow subgroups of finite groups.
Let \(P\) be a Sylow \(p\)-subgroup of a finite group \(G\) and let \(b(G)\) denote the largest degree of an irreducible complex character of \(G\). In the paper under review the authors show: For any finite group \(G\) the following bounds hold: a) \(|P/O_p(G)|<b(G)^2\). b) \(|P/O_p(G)|\leq b(G)\) if \(P\) is Abelian. The proof splits into two parts. Part one, which deals with the case that all minimal normal subgroups of \(G\) are solvable, is mainly based on a paper of \textit{I. M. Isaacs} [Proc. Am. Math. Soc. 127, No. 1, 45-50 (1999; Zbl 0916.20011)]. The other part, in which \(G\) has a non-solvable minimal normal subgroup, uses the classification of finite non-Abelian simple groups \(H\) and a careful analysis of the largest degree of \(H\) in relation to the order of the \(p\)-part of \(|\Aut(H)|\).
scientific article; zbMATH DE number 2102473