Characterizing Groups By The Sum Of Element Orders And The Maximal Order

It is well known that finite simple groups are basis of finite groups. In order to know better their properties and structures, it is very helpful for us to characterize them by their obvious, simple and instinctive properties.In this paper, we will study the characterization of finite simple groups by using the sum of element orders and the maximal order of elements. In this paper, we use m(G) to denote the maximal order of elements and ψ(G) to denote the sum of element orders. We get the following results:Theorem 3.4 we can characterize A5 by m(G) and ψ(G), where A5 is the minimal order of non-Abelian simple group, i.e.G(?) A5 if and only if ψ(G)= ψ(A5)= 211 and m(G)= m(A5)= 5.Theorem 4.8 we can characterize PSL(2,7) by m(G) and ψ(G), where PSL(2,7) is the sub-minimal order of non-Abelian simple group, i.e.G(?) PST(2,7) if and only if ψ(G)= ψ(PSL(2,7))= 715 and = m(PSL(2,7))= 7.