The largest character degree and the Sylow subgroups of finite groups. (Q2803550)

In this paper the following result is proved. Let \(p\) be a prime and \(P\) a Sylow \(p\)-subgroup and \(b(G)\) the largest irreducible character degree of a finite nonabelian group \(G\). Then \(|P/O_p(G)|\leq (b(G)^p/p)^{\tfrac{1}{p-1}}\). This bound had been obtained for \(p\)-solvable groups by \textit{M. L. Lewis} [in Commun. Algebra 42, No. 5, 1994-2002 (2014; Zbl 1300.20010)]; the weaker bound \(b(G)^2\) had been known since 2004 by work of the first author and \textit{W. Shi} [J. Algebra 277, No. 1, 165-171 (2004; Zbl 1062.20008)].