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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]

Motivated by Isaacs and Passman's characterization of finite groups all of whose nonlinear complex irreducible characters have prime degrees, we investigate finite groups with exactly one character degree that is not a prime. We show that either is solvable with or for distinct primes , or up to an abelian direct factor, is isomorphic to the alternating group . [ABSTRACT FROM AUTHOR]