Hall subgroups need not exist
Let be a finite group and be a set of prime numbers. Then, there need not exist a -Hall subgroup (?) of . In other words, there need not exist a subgroup whose order and index are relatively prime, and where all prime factors of the order are in .
Facts about Sylow subgroups
More facts about existence of Hall subgroups
- Hall subgroups exist in finite solvable
- Hall's theorem on solvability: This states that Hall subgroups of all possible orders exist for a finite group if and only if the finite group is a finite solvable group.
Proof using Hall's theorem
The proof directly follows from Hall's theorem, and the fact that there exist finite groups that are not solvable; for instance, the alternating group of degree five.
A concrete example
Further information: Alternating group:A5
Let be the alternating group of degree five. Consider the prime set . For a -Hall subgroup to exist, it must have order equal to . However, has no subgroup of order .
There are many ways of seeing this. For instance, let's assume we know that A5 is simple. Then, a subgroup of order gives an action of on the coset space of that subgroup (which has size three), yielding a nontrivial homomorphism from to the symmetric group on three elements. This contradicts the simplicity of .