a criterion for p-solvability of finite groups, where p = 7 or 11

For a finite group g, define ψ´´(g) = ψ(g)/|g|², where ψ(g) = σg∈go(g) and o(g) denotes the order of g ∈ g. in this paper, we give a criterion for p-solvability by the function ψ´´, where p ∈ {7, 11}. we prove that if g is a finite group and ψ´´(g) > ψ´´(psl(2, p)), where p ∈ {7, 11}, then g is a p-solvable group.