University of Isfahan International Journal of Group Theory 2251-7650 11 1 2022 03 01 The recognition of finite simple groups with no elements of order $10$ by their element orders 17 22 25569 10.22108/ijgt.2021.124142.1640 EN Huaiyu He Department of Economics and Management, Shanghai University of Political Science and Law, Shanghai 201701, China Wujie Shi Department of Mathematics, Chongqing University of Arts and Sciences, Chongqing 402160, China3School of Mathe- matics, Suzhou University, Suzhou 215006, China Journal Article 2020 07 25 The spectrum of a finite group is the set of‎ ‎its element orders‎. ‎$H$ is said to be a finite cover of $G$ if $G$‎ ‎is a homomorphic image of $H$ and $H$ is finite‎. ‎The main aim of‎ ‎this article is to characterize the finite simple groups with no‎ ‎elements of order 10 by its spectrum among covers‎. ‎At the same‎ ‎time‎, ‎above simple groups are completely classified‎. ‎At last‎, ‎some‎ ‎results on the recognition by spectrum of above groups are also‎ ‎achieved‎. https://ijgt.ui.ac.ir/article_25569_756a080256c4fec90519527cd91aec44.pdf