On Some Conjectures Related to Quantitative Characterizations of Finite Nonabelian Simple Groups

This paper is based on the results of the 2020 Ural Workshop on Group Theory and Combinatorics. In this note we provide some counterexamples for the conjecture of Moretó on finite simple groups, which says that any finite simple group $G$ can be determined in terms of its order $|G|$ and the number of elements of order $p$, where $p$ the largest prime divisor of $|G|$. A new characterization of all sporadic simple groups and alternating groups is given. Some related conjectures are also discussed.