20.27 (2022)

Let $G$ be a finite group, $p$ a prime number, and let $|g^G|_p$ denote the maximum power of $p$ that divides the class size of an element $x \in G$. Suppose that there exists a $p$-element $g \in G$ such that $|g^G|_p = \max_{x \in G} |x^G|_p$. Is it true that $G$ has a normal $p$-complement?