Quantitative Characterization Of Finite Simple Groups

Let G be a finite group. Denote by xe(G) the set of all orders of elements in G, by h(ne(G)) the number of isomorphism classes of groups H such that ite(H) = ne(G), and by \G\ the order of G. In this paper we prove the following three theorems:Theorem A Let G be a group, H = Um(q), q=p",pisa prime. Then G = Hif and only if (1) e(G) = ne(H), (2) \G\ = \H\.Theorem B Let G be a finite group, L = L3(2m), m>1,orL = U3(2m), m>2.If ne(G) = ne(L), then G = L.Theorem C Let H be the exceptional Chevalley groups G2(q), or the twisted groups 3D4(q), where q=pn,pisa prime, then h( e(H)) {1, }.