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In [[mathematics]], specifically [[group theory]], a '''Hall subgroup''' of a [[finite group]] ''G'' is a [[subgroup]] whose [[order (group theory)|order]] is [[coprime]] to its [[Index of a subgroup|index]]. They were introduced by the group theorist {{harvs|txt|authorlink=Philip Hall|first=Philip|last=Hall|year=1928}}.

== Definitions ==

A '''Hall divisor''' (also called a [[unitary 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 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 2<sup>2</sup>&thinsp;×&thinsp;3&thinsp;×&thinsp;5, so one takes any product of 3, 2<sup>2</sup>&thinsp;=&thinsp;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 [[prime number|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 (mathematics)|group]] is a Hall subgroup.
*The [[alternating group]] ''A''<sub>4</sub> of order 12 is [[solvable group|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''<sub>5</sub>, the only [[simple group]] 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_class#Conjugacy_of_subgroups_and_general_subsets|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 [[group isomorphism|isomorphic]] (and so certainly not [[conjugate subgroups|conjugate]], even under an outer automorphism). The [[normalizer]] of a Sylow {{nowrap|2-subgroup}} of order 4 is isomorphic to the alternating group ''A''<sub>4</sub> of order 12, while the normalizer of a subgroup of order 2 or 3 is isomorphic to the [[dihedral group]] of order 12.

== Hall's theorem ==

{{harvtxt|Hall|1928}} [[mathematical proof|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 {{nowrap|''π''-subgroups}} are conjugate. Moreover, any subgroup whose order is
a product of primes in ''π'' is contained in some Hall {{nowrap|''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 [[mathematical induction|induction]] on the order of ''G'', using the fact that every finite solvable group has a [[normal subgroup|normal]] [[elementary abelian]] subgroup. More precisely, fix a [[minimal normal subgroup]] ''A'', which is either a {{nowrap|''π''-group}} or a {{nowrap|''π′''-group}} as ''G'' is {{nowrap|''π''-separable}}. By induction there is a subgroup ''H'' of ''G'' containing ''A'' such that ''H''/''A'' is a Hall {{nowrap|''π''-subgroup}} of ''G''/''A''. If ''A'' is a {{nowrap|''π''-group}} then ''H'' is a Hall {{nowrap|''π''-subgroup}} of ''G''.  On the other hand, if ''A'' is a {{nowrap|''π′''-group}}, then by the [[Schur–Zassenhaus theorem]] ''A'' has a [[complement (group theory)|complement]] in ''H'', which is a Hall {{nowrap|''π''-subgroup}} of ''G''.

== A converse to Hall's theorem ==

Any finite group that has a Hall {{nowrap|''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<sup>a</sup>q<sup>b</sup>'' 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 (logic)|converse]].

== Sylow systems ==

A '''Sylow system''' is a set of Sylow {{nowrap|''p''-subgroups}} ''S<sub>p</sub>'' for each prime ''p'' such that ''S<sub>p</sub>S<sub>q</sub>'' = ''S<sub>q</sub>S<sub>p</sub>'' for all ''p'' and ''q''. If we have a Sylow system, then the subgroup generated by the groups ''S<sub>p</sub>'' for ''p'' in ''π'' is a Hall {{nowrap|''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 normal Hall subgroup ''H'' of a finite group ''G'' possesses a [[complement (group theory)|complement]], that is, there is some subgroup ''K'' of ''G'' that [[intersection (set theory)|intersects]] ''H'' trivially and such that ''HK''&nbsp;=&nbsp;''G'' (so ''G'' is a [[semidirect product]] of ''H'' and ''K''). This is the [[Schur–Zassenhaus theorem]].

==See also==

*[[formation (group theory)|Formation]]

== References ==
*{{citation|title=Finite groups|first=Daniel|last=Gorenstein|author-link=Daniel Gorenstein| isbn =0-8284-0301-5|year=1980|publisher=Chelsea Publishing Co.|location=New York|mr=0569209}}.
*{{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}}

[[Category:Finite groups]]
[[Category:Solvable groups]]
[[Category:Subgroup properties]]