Quantitative Characterization of Finite Simple Groups and Related Topics

For a finite group, the order of group and the orders are two most important basic concepts. Let $G$ be a finite group and $\pi_e(G)$ be the set of element orders in $G$. In 1987, we posed the following conjecture. Let $G$ be a group and $M$ a finite simple group. Then $G\cong M$ if and only if (a) $\pi_e(G) = \pi_e(M)$, and (b) $|G| = |M|$. That is, for all finite simple groups we may characterize them using only their orders and the sets of their element orders. Now the above conjecture has became a theorem after a series of proofs by Chinese and Russian mathematicians (1987-2009). In this talk, we will discuss the above characterization and related topics. Especially, some unsolved problems depending on number theory and Diophantine equations will be discussed.