The Strong -Sylow Theorem for the Groups PSL

Let be a set of primes. A finite group is a -group if all prime divisors of the order of belong to . Following Wielandt, the -Sylow theorem holds for if all maximal -subgroups of are conjugate; if the -Sylow theorem holds for every subgroup of then the strong -Sylow theorem holds for . The strong -Sylow theorem is known to hold for if and only if it holds for every nonabelian composition factor of . In 1979, Wielandt asked which finite simple nonabelian groups obey the strong -Sylow theorem. By now the answer is known for sporadic and alternating groups. We give some arithmetic criterion for the validity of the strong -Sylow theorem for the groups .