Finite Groups Determined By The Sum Of Its Element Orders

Given a finite group G of order n, let ψ(G)=Σx∈Go (x), where o(x) is the order of the element x. In [3], H.Amiri and I.M.Isaacs proved that in all groups of order n, max{ψ(G)||(G)|=n)=ψ(Zn), where Zn is the cyclic group of order n. For any group H of order n, which is not isomorphic to G and Zn, ifψ(H)<ψ(G)<ψ(Zn), we say that ψ(G) is the second largest in all groups of order n. In this thesis, we consider the structure of finite groups whose sum of element orders is the second largest between all groups of the same order. Let p be a prime, n be a natural number, n>≥2, and when p=2, n≥3,|G|=pn+1.We first prove that ifψ(H)<ψ(G)<ψ(Zpn+1) for any group H of order pn+1not isomorphic to G, then G must be isomorphic to Zpn x Zp or Mpn+1,where Mpn+1=(a,b|apn=1,bp=1, b-1ab=a1+pn-1). Then we obtain the structure of the abel group whose sum of element orders is the second largest between all groups of order n. At last,we obtain some properties of finite group whose sum of element orders is the second largest between all groups of order2n+1pk(p≠2) and2n+13, namely, it must have a cyclic normal subgroup of index2.