Influence of the number of Sylow subgroups on solvability of finite groups

Abstract

Let G be a finite group. We prove that if the number of Sylow 3-subgroups of G is at most 7 and the number of Sylow 5-subgroups of G is at most 1455, then G is solvable. This is a strong form of a recent conjecture of Robati. 2020 Mathematics Subject Classification. 20D10, 20D20, 20F16, 20F19. Funding. The first author is supported by both TU Graz (R-1501000001) and partial funding from the Austrian Science Fund (FWF): P30934–N35, F05503, F05510. He is also at the University of Nigeria, Nsukka (UNN). The research of the second author is supported by Ministerio de Ciencia e Innovación PID−2019−103854GB−100, Generalitat Valenciana AICO/2020/298 and FEDER funds. Manuscript received 4th October 2020, revised and accepted 5th November 2020.