Covering the set of $p$-elements in finite groups by Sylow $p$-subgroups

Attila Maróti, Juan Martínez, Alexander Moretó

Let $G_p$ be the set of $p$-elements of a finite group $G$. Do we need all the Sylow $p$-subgroups of $G$ to cover $G_p$? Although this question does not have an affirmative answer…