THE RECOGNITION OF FINITE SIMPLE GROUPS WITH NO ELEMENTS OF ORDER 10 BY THEIR ELEMENT ORDERS

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.