A criterion for p-nilpotency and p-closedness by the sum of element orders

摘要

LetGbe a finite group,psi(G)= n-ary sumation g is an element of Go(g)and psi ''(G)=psi(G)/|G|2,whereo(g) denotes the order ofg. In this paper, we prove that ifpis a prime number and psi ''(G)>(p2+p+1)/(4p2),thenG=Op(G)xOp '(G),whereOp(G)is cyclic. In addition,Gis supersolvable andG ''<= Z(G).