quantitative characterization of finite simple groups: a complement

In this paper, we summarize the research on the characterization of finite simple groups and the study of finite groups based on their “set of element orders” and “two orders” (the order of the group and the set of element orders). we also discuss some related topics, their applications, and generalizations. the original version of this paper was published in chinese in [scientia sinica mathematica, 53 (2023), no. 7, 931–952]. this updated version corrects several errors and includes additional content. it particularly emphasizes the applications of this work in mathematics and computational complexity theory.