# Hall subgroup

Let $G$ be a finite group. A subgroup $H$ of $G$ is said to be a Hall subgroup if

 $\gcd(|H|,|G/H|)=1.$

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 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 $p^{k}m$, $\gcd(p,m)=1$, has a Hall subgroup (of order $p^{k}$).

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

###### Theorem 1 (Hall (1928)).

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

The sets of primes $\pi$ 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 $A_{5}$ has no Hall $\{2,5\}$-subgroup. That is, $A_{5}$ has no subgroup of order $20$.

###### Proof.

Suppose that $A_{5}$ has a Hall $\{2,5\}$-subgroup $H$. As $|A_{5}|=60$, 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 $A_{5}$ acts on the three member set $C$ of cosets of $H$. This induces a non-trivial homomorphism from $A_{5}$ to $S_{C}\cong S_{3}$ (here, $S_{C}$ is the symmetric group on $C$, see this (http://planetmath.org/GroupActionsAndHomomorphisms) for more detail). Since $A_{5}$ 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 $A_{5}$. In any case, $A_{5}$ is non-abelian simple and therefore it is not a solvable group. Thus, Hall’s theorem does not apply to $A_{5}$.

Title Hall subgroup HallSubgroup 2013-03-22 14:02:02 2013-03-22 14:02:02 Algeboy (12884) Algeboy (12884) 22 Algeboy (12884) Definition msc 20D20 VeeCuhininsTheorem SylowTheorems Hall’s theorem Hall $\pi$-subgroup