ON THE CHARACTERIZABILITY OF SOME FAMILIES OF FINITE GROUP OF LIE TYPE BY ORDERS AND VANISHING ELEMENT ORDERS

Abstract

In this paper, we show that the following simple groups are uniquely determined by their orders and vanishing element orders: Ap-1(2), where p ̸= 3, 2Dp+1(2),where p ≥ 5, p ̸= 2m - 1, Ap(2), Cp(2), Dp(2), Dp+1(2) which for all of them p is anodd prime and 2p - 1 is a Mersenne prime. Also, 2Dn(2) where 2n-1 + 1 is a Fermatprime and n > 3, 2Dn(2) and Cn(2) where 2n + 1 is a Fermat prime. Then we give analmost general result to recognize the non-solvability of finite group H by an anologybetween orders and vanishing elemen orders of H and a finite simple group of Lie type.