Hall subgroup

Let G be a finite groupMathworldPlanetmath. A subgroupMathworldPlanetmathPlanetmath H of G is said to be a Hall subgroup if


In other words, H is a Hall subgroup if the order of H and its index in G are coprime. These subgroups are name after Philip Hall who used them to characterize solvable groupsMathworldPlanetmath.

Hall subgroups are a generalizationPlanetmathPlanetmath of Sylow subgroups. Indeed, every Sylow subgroup is a Hall subgroup. According to Sylow’s theoremMathworldPlanetmath, this means that any group of order pkm, gcd(p,m)=1, has a Hall subgroup (of order pk).

A common notation used with Hall subgroups is to use the notion of π-groups (http://planetmath.org/PiGroupsAndPiGroups). Here π is a set of primes and a Hall π-subgroup of a group is a subgroup which is also a π-group, and maximal with this property.

Theorem 1 (Hall (1928)).

A finite group G is solvable iff G has a Hall π-subgroup for any set of primes π.

The sets of primes π in Hall’s theorem can be restricted to the subsets of primes which divide |G|. However, this result fails for non-solvable groups.

Example 2.

The group A5 has no Hall {2,5}-subgroup. That is, A5 has no subgroup of order 20.


Suppose that A5 has a Hall {2,5}-subgroup H. As |A5|=60, it follows that |H|=20. Thus, there are three cosets of H. Since a group always acts on the cosets of a subgroup, it follows that A5 acts on the three member set C of cosets of H. This induces a non-trivial homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath from A5 to SCS3 (here, SC is the symmetric groupPlanetmathPlanetmath on C, see this (http://planetmath.org/GroupActionsAndHomomorphisms) for more detail). Since A5 is simple, this homomorphism must be one-to-one, implying that its image must have order at most 6, an impossibility. ∎

This example can also be proved by direct inspection of the subgroups of A5. In any case, A5 is non-abelianMathworldPlanetmathPlanetmath simple and therefore it is not a solvable group. Thus, Hall’s theorem does not apply to A5.

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 π-subgroup