On The Largest Character Degree And Solvable Subgroups Of Finite Groups

Let $G$ be a finite group, and $\pi$ be a set of primes. The $\pi$-core ${\mathbf{O}}_{\pi }(G)$ is the unique maximal normal $\pi$-subgroup of $G$, and $b(G)$ is the largest irreducible character degree of $G$. In 2017, Qian and Yang proved that if $H$ is a solvable $\pi$-subgroup of $G$, then $\mathrm{\mid }H{\mathbf{O}}_{\pi }(G)\mathrm{/}{\mathbf{O}}_{\pi }(G)\mathrm{\mid }\le b(G{)}^3$. In this paper, we improve the exponent of $3$ to $3{\log }_{504}(168)<2.471$.