p-solvable finite groups, under special prime p = 11

Let g be a finite group, and let m(g) = ∑g2g 1=o(g), where o(g) denotes the order of g. in this paper, we determine all groups such that m(g) < 4. we provide a criterion for p-solvability using the function m(g), where p 2 f11g then g is a p-solvable group.