Hall subgroup

In mathematics, a Hall subgroup of a finite group G is a subgroup whose order is coprime to its index. They were introduced by the group theorist Philip Hall (1928).

A Hall subgroup of G is a subgroup whose order is a Hall divisor of the order of G. In other words, it is a subgroup whose order is coprime to its index.

If π is a set of primes, then a Hall π-subgroup is a subgroup whose order is a product of primes in π, and whose index is not divisible by any primes in π.

Hall (1928) proved that if G is a finite solvable group and π is any set of primes, then G has a Hall π-subgroup, and any two Hall π-subgroups are conjugate. Moreover, any subgroup whose order is a product of primes in π is contained in some Hall π-subgroup. 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 π-group or a π'-group as G is π-separable. By induction there is a subgroup H of G containing A such that H/A is a Hall π-subgroup of G/A. If A is a π-group then H is a Hall π-subgroup of G. On the other hand, if A is a π'-group, then by the Schur–Zassenhaus theorem A has a complement in H, which is a Hall π-subgroup of G.

Any finite group that has a Hall π-subgroup 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 p aq b 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.

A Sylow system is a set of Sylow p-subgroups 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 π-subgroup. A more precise version of Hall's theorem says that any solvable group has a Sylow system, and any two Sylow systems are conjugate.

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.