mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, 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 Finite is the opposite of infinite. It may refer to: * Finite number (disambiguation) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked ...

''G'' is a

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...

whose

order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of d ...

coprime In mathematics, 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 equivale ...

to its

index Index (or its plural form 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 a Halo megastru ...

. 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. Thus, 5 is a unitary divisor of 60, because 5 an ...

) of an

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

''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 divisible by ...

''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 In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: , and are prime powers, while , and are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17 ...

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 A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

, 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 divisible by ...

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 fixed ...

of a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

is a Hall subgroup. *The

alternating group In mathematics, an alternating group is the 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 by or Basic prop ...

''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 In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other wor ...

of Hall subgroups of order 24 (though they are connected by an

outer automorphism In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has a t ...

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 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 them. The word is ...

(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 of elements \mathrm_G(S) of ''G'' such that each member g \in \mathrm_G(S) commutes with each element of ''S'', o ...

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 of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ge ...

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 terminates ...

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 Induction, Inducible or Inductive may refer to: Biology and medicine * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...

on the order of ''G'', using the fact that every finite solvable group has a

normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...

elementary abelian In mathematics, specifically in group theory, an elementary abelian group (or elementary abelian ''p''-group) is an abelian group in which every nontrivial element has order ''p''. The number ''p'' must be prime, and the elementary abelian grou ...

subgroup. More precisely, fix a

minimal normal subgroup In mathematics, in the field of group theory, a group is said to be characteristically simple if it has no proper nontrivial characteristic subgroups. Characteristically simple groups are sometimes also termed elementary groups. Characteristically s ...

''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 The Schur–Zassenhaus theorem is a theorem in group theory which states that if G is a finite group, and N is a normal subgroup whose order is coprime to the order of the quotient group G/N, then G is a semidirect product (or split extension ...

''A'' has a

complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...

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 In mathematics, Burnside's theorem in group theory states that if ''G'' is a finite group of order p^a q^b where ''p'' and ''q'' are prime numbers, and ''a'' and ''b'' are non-negative integers, then ''G'' is solvable. Hence each non-Abelian fin ...

that any group whose order is of the form ''p^aq^b'' for primes ''p'' and ''q'' is solvable, because

Sylow's theorem 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 fixed ...

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 Converse may refer to: Mathematics and logic * Converse (logic), the result of reversing the two parts of a definite or implicational statement ** Converse implication, the converse of a material implication ** Converse nonimplication, a logical c ...

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

, 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. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in w ...

of ''H'' and ''K''). This is the

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 The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical S ...

, 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