University of Isfahan International Journal of Group Theory 2251-7650 14 4 2024 11 01 Quantitative characterization of finite simple groups: a complement 181 222 28685 10.22108/ijgt.2024.140804.1893 EN Wujie Shi School of Mathematical Sciences, Suzhou University Suzhou, Jiangsu, P. R. China and School of Mathematics and Big Date, Chongqing University of Arts and Sciences, Chongqing, P. R. China Journal Article 2024 01 24 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, <strong>53</strong> (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. Finite group Group orders element orders Classification theorem for finite simple groups https://ijgt.ui.ac.ir/article_28685_0ded34e0ab5cc4a7371098f9030eae4f.pdf