The effect of the sum of the inverse-power of element orders on finite groups

Let G be a finite group, and let m(G)=\sum_{g \in G}۱/o(g), where o(g) denotes the order of g. In this paper, we determine all groups such that m(G)<۴. Additionally, for a finite group G of odd order, we investigate the influence of m(G) on supersolvability. Finally, we provide a criterion for p-solvability using the function m(G), where p \in \{۷,۱۱\}.

کلیدواژه ها:

p-solvability ، Sum of the inverse-power of element orders ، Supersolvability