In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Hall subgroup of a finite group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 G is a subgroup whose order is coprime

Coprime

In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...

 to its index

Index of a subgroup

In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...

. They are named after the group theorist Philip Hall

Philip Hall

Philip Hall FRS , was an English mathematician.His major work was on group theory, notably on finite groups and solvable groups.-Biography:...

.

Definitions

A Hall divisor of an integer n is a divisor d of n such that
d and n/d are coprime. The easiest way to find the Hall divisors is to write the prime factorization for the number in question and take any product of the multiplicative terms (the full power of any of the prime factors), including 0 of them for a product of 1 or all of them for a product equal to the original number. For example, to find the Hall divisors of 60, show the prime factorization is 2²·3·5 and take any product of {3,4,5}. Thus, the Hall divisors of 60 are 1, 3, 4, 5, 12, 15, 20, and 60.

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.

Examples

• Any Sylow subgroup of a group is a Hall subgroup.

• If G = A₅, the only simple group
  Simple group
  In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
  
   of order 60, then 15 and 20 are Hall divisors of the order of G, but G has no subgroups of these orders.

• The simple group of order 168 has two different conjugacy classes of Hall subgroups of order 24 (though they are conjugate under an outer automorphism of G).

• The simple group of order 660 has two Hall subgroups of order 12 that are not isomorphic.

Hall's theorem

Hall proved that if G is a finite solvable group

Solvable group

In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...

 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.

Hall's theorem can be proved by induction on the order of G, using the fact that every finite solvable group has a normal elementary abelian subgroup.

A converse to Hall's theorem

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 give another proof of Burnside's theorem, because Burnside's theorem is needed to prove this converse.

Sylow systems

A Sylow system is a set of Sylow p-subgroups S_p for each prime p such that S_pS_q = S_qS_p for all p and q. If we have a Sylow system, then the subgroup generated by the groups S_p 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.

Normal Hall subgroups

Any Hall normal Hall subgroup H of a finite group G possesses a complement, that is there is some subgroup K of G which intersects H trivially and such that HK=G (so G is isomorphic to a semi-direct product of H and K).