SOME CRITERIA FOR SOLVABILITY AND SUPERSOLVABILITY

Denote by $ G $ a finite group, by  $ {\rm hsn}(G) $ the harmonic mean Sylow number (eliminating the Sylow numbers that are one) in $G$ and by    $ {\rm gsn}(G) $ the geometric mean Sylow number (eliminating the Sylow numbers that are one) in $G$. In this paper, we prove that if either $ {\rm hsn}(G) 45/7 $ or  $ {\rm gsn}(G) \sqrt[3]{300} $, then $G$ is solvable. Also, we show that if either $ {\rm hsn}(G) 24/7 $ or  $ {\rm gsn}(G) \sqrt{12} $, then $G$ is supersolvable.