Mersenne primes and solvable Sylow numbers.

Let be a prime. The Sylow -number of a finite group , which is the number of Sylow -subgroups of , is called solvable if its -part is congruent to modulo for any prime . P. Hall showed that solvable groups only have solvable Sylow numbers, and M. Hall showed that the Sylow -number of a finite group is the product of two kinds of factors: of prime powers with (mod ) and of the number of Sylow -subgroups in certain finite simple groups (involved in ). These classical results lead to the investigation of solvable Sylow numbers of finite simple groups. In this paper, we show that a finite nonabelian simple group has only solvable Sylow numbers if and only if it is isomorphic to for a Mersenne prime. [ABSTRACT FROM AUTHOR]