A bound for $|G:{\mathbf{O}}_{p}(G)|_{p}$ in terms of the largest irreducible character degree of a finite $p$-solvable group $G$

Let $b(G)$ denote the largest irreducible character degree of a finite group $G$, and let $p$ be a prime. Two results are obtained. First, we show that, if $G$ is a $p$-solvable group and if $b(G) < p^{2}$, then $p^{2} {\not \big \vert }\,|\,G:{\mathbf{O}}_{p}(G)|$. Next, we restrict attention to solvable groups and show that, if $b(G) \le p^{\alpha }$ and if $P$ is a Sylow $p$-subgroup of $G$, then $|P: {\mathbf{O}}_{p}(G)|\le p^{2\alpha }$.