Hall subgroup
Let be a finite group. A subgroup
of is said to
be a Hall subgroup if
In other
words, is a Hall subgroup if the order of and its index
in are coprime. These subgroups are
name after Philip Hall who used them to characterize solvable groups.
Hall subgroups are a generalization of Sylow subgroups. Indeed,
every Sylow subgroup is a Hall subgroup. According to Sylow’s
theorem
, this means that any group of order , ,
has a Hall subgroup (of order ).
A common notation used with Hall subgroups is to use the notion of -groups (http://planetmath.org/PiGroupsAndPiGroups). Here is a set of primes and a Hall -subgroup of a group is a subgroup which is also a -group, and maximal with this property.
Theorem 1 (Hall (1928)).
A finite group is solvable iff has a Hall -subgroup for any set of primes .
The sets of primes in Hall’s theorem can be restricted to the subsets of primes which divide . However, this result fails for non-solvable groups.
Example 2.
The group has no Hall -subgroup. That is, has no subgroup of order .
Proof.
Suppose that has a Hall -subgroup .
As , it follows that . Thus, there
are three cosets of . Since a group always acts
on the cosets of a subgroup, it follows that acts
on the three member set of cosets of . This induces
a non-trivial homomorphism from to
(here, is the symmetric group
on , see
this (http://planetmath.org/GroupActionsAndHomomorphisms) for more detail).
Since is simple, this homomorphism must be one-to-one,
implying that its image must have order at most , an
impossibility.
∎
This example can also be proved by direct inspection of the subgroups of .
In any case, is non-abelian simple and therefore it is not a
solvable group. Thus, Hall’s theorem does not apply to .
Title | Hall subgroup |
---|---|
Canonical name | HallSubgroup |
Date of creation | 2013-03-22 14:02:02 |
Last modified on | 2013-03-22 14:02:02 |
Owner | Algeboy (12884) |
Last modified by | Algeboy (12884) |
Numerical id | 22 |
Author | Algeboy (12884) |
Entry type | Definition |
Classification | msc 20D20 |
Related topic | VeeCuhininsTheorem |
Related topic | SylowTheorems |
Defines | Hall’s theorem |
Defines | Hall -subgroup |