  In this paper, we summarize the work on the characterization of finite simple
groups and the study on finite groups with the set of element orders and two
orders (the order of group and the set of element orders). Some related topics,
and the applications together with their generalizations are also discussed.
The original version of this article was published in Chinese in the journal
Scientia Sinica Mathematica, no.53(2023), pp.931-952. This revised and expanded
version has corrected several errors and added quite a few contents.
Especially, it is pointed that this work has applications in mathematics and
computational complexity theory.
