Some criteria for solvability and supersolvability

Abstract

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.

Keywords

• finite group
• sylow subgroup
• solvable groups