Hall are conjugate. Moreover, any subgroup whose order is a product of primes in π is contained in some Hall . This result can be thought of as a generalization of Sylow's Theorem to Hall subgroups, but the examples above show that such a generalization is false when the group is not solvable. The existence of Hall subgroups can be proved by induction on the order of G, using the fact that every finite solvable group has a normal elementary abelian subgroup. More precisely, fix a minimal normal subgroup A, which is either a or a as G is . By induction there is a subgroup H of G containing A such that H/A is a Hall of G/A. If A is a then H is a Hall of G. On the other hand, if A is a , then by the Schur–Zassenhaus theorem A has a complement in H, which is a Hall of G. A converse to Hall's theorem Any finite group that has a Hall for every set of primes π is solvable. This is a generalization of Burnside's theorem that any group whose order is of the form paqb for primes p and q is solvable, because Sylow's theorem implies that all Hall subgroups exist. This does not (at present) give another proof of Burnside's theorem, because Burnside's theorem is used to prove this converse. Sylow systems A Sylow system is a set of Sylow Sp for each prime p such that SpSq = SqSp for all p and q. If we have a Sylow system, then the subgroup generated by the groups Sp for p in π is a Hall . A more precise version of Hall's theorem says that any solvable group has a Sylow system, and any two Sylow systems are conjugate. Normal Hall subgroups Any normal Hall subgroup H of a finite group G possesses a complement, that is, there is some subgroup K of G that intersects H trivially and such that HK = G (so G is a semidirect product of H and K). This is the Schur–Zassenhaus theorem. See also Formation References . Finite groups Solvable groups Subgroup properties