  In this note we provide some counterexamples for the conjecture of Moret\'{o}
on finite simple groups, which says that any finite simple group $G$ can
determined in terms of its order $|G|$ and the number of elements of order $p$,
where $p$ the largest prime divisor of $|G|$. Moreover, we show that this
conjecture holds for all sporadic simple groups and alternating groups $A_n$,
where $n\neq 8, 10$. Some related conjectures are also discussed.
