4.21 (1973)

Let $G$ be a finite group, $p$ an odd prime number, and $P$ a Sylow $p$-subgroup of $G$. Let the order of every non-identity normal subgroup of $G$ be divisible by $p$. Suppose $P$ has an element $x$ that is conjugate to no other from $P$. Does $x$ belong to the centre of $G$? For $p = 2$, the answer is positive (G. Glauberman, J. Algebra, 4 (1966), 403–420).