mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, specifically

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

, a Hall subgroup of a

finite group In abstract algebra, a finite group is a group whose underlying set is finite. Finite groups often arise when considering symmetry of mathematical or physical objects, when those objects admit just a finite number of structure-preserving tra ...

''G'' is a

subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

whose order is

coprime In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...

to its

index Index (: indexes or indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on the Halo Array in the ...

. They were introduced by the group theorist .

Definitions

A Hall divisor (also called a

unitary divisor In mathematics, a natural number ''a'' is a unitary divisor (or Hall divisor) of a number ''b'' if ''a'' is a divisor of ''b'' and if ''a'' and \frac are coprime, having no common factor other than 1. Equivalently, a divisor ''a'' of ''b'' is a un ...

) of an

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

''n'' is a

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...

''d'' of ''n'' such that ''d'' and ''n''/''d'' are coprime. The easiest way to find the Hall divisors is to write the prime power factorization of the number in question and take any subset of the factors. For example, to find the Hall divisors of 60, its prime power factorization is 2² × 3 × 5, so one takes any product of 3, 2² = 4, and 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 In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...

by any primes in ''π''.

Examples

*Any

Sylow subgroup In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixe ...

of a group is a Hall subgroup. *The

alternating group In mathematics, an alternating group is the Group (mathematics), group of even permutations of a finite set. The alternating group on a set of elements is called the alternating group of degree , or the alternating group on letters and denoted ...

''A''₄ of order 12 is solvable but has no subgroups of order 6 even though 6 divides 12, showing that Hall's theorem (see below) cannot be extended to all divisors of the order of a solvable group. *If ''G'' = ''A''₅, the only

simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...

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 connected by an outer automorphism of ''G''). *The simple group of order 660 has two Hall subgroups of order 12 that are not even

isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

(and so certainly not conjugate, even under an outer automorphism). The

normalizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of ele ...

of a Sylow of order 4 is isomorphic to the alternating group ''A''₄ of order 12, while the normalizer of a subgroup of order 2 or 3 is isomorphic to the

dihedral group In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...

of order 12.

Hall's theorem

proved that if ''G'' is a finite

solvable group In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminat ...

and ''π'' is any set of primes, then ''G'' has a Hall ''π''-subgroup, and any two 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 ''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.

Sylow systems

A Sylow system is a set of Sylow ''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 . 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 In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol . There are two closely related concepts of semidirect product: * an ''inner'' sem ...

of ''H'' and ''K''). This is the Schur–Zassenhaus theorem.

References

*. *{{citation, last=Hall, first= Philip, author-link=Philip Hall , title=A note on soluble groups, jfm= 54.0145.01 , journal= Journal of the London Mathematical Society , volume=3, issue= 2 , pages= 98–105 , year=1928, doi=10.1112/jlms/s1-3.2.98, mr=1574393 Finite groups Solvable groups Subgroup properties