Hall's Theorem
Hall 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 π-subgroups are conjugate. Moreover any subgroup whose order is a product of primes in π is contained in some Hall 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.
Hall's theorem can be proved by induction on the order of G, using the fact that every finite solvable group has a normal elementary abelian subgroup.
Read more about this topic: Hall Subgroup
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