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)<۴۵/۷ or   {\rm gsn}(G)< \sqrt[۳]{۳۰۰} , then G is solvable. Also, we show that if either {\rm hsn}(G)<۲۴/۷ or   {\rm gsn}(G)<\sqrt{۱۲} , then G is supersolvable.

Keywords:

Finite group , Sylow subgroup , solvable groups