# 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 ^{ a}q^{ 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 *S _{p}* for each prime

*p*such that

*S*=

_{p}S_{q}*S*for all

_{q}S_{p}*p*and

*q*. If we have a Sylow system, then the subgroup generated by the groups

*S*for

_{p}*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.