Def. Let be a finite group. A subgroup of is said to be a Hall subgroup if$$ \gcd(|H|,|G/H|)=1.
**Exm.** Every Sylow subgroup is a Hall subgroup. A common notation used with Hall subgroups is to use the notion of $\pi$-groups. Here $\pi$ is a set of primes and a Hall $\pi$-subgroup of a group is a subgroup which is also a $\pi$-group. **Thm.** *(Hall)* A finite group $G$ is solvable iff $G$ has a Hall $\pi$-subgroup for any set of primes $\pi$. **Exm.** The group $A_5$ has no Hall $\{2,5\}$-subgroup and so $A_5$ is not solvable.