The Sylow theorems are important tools for analysis of special subgroups of a finite group known as Sylow subgroups. They are especially useful in the classification of finite simple groups.
The first Sylow theorem guarantees the existence of a Sylow subgroup of for any prime dividing the order of A Sylow subgroup is a subgroup whose order is a power of and whose index is relatively prime to The second Sylow theorem states that all the Sylow subgroups of a given order are conjugate, and the third Sylow theorem gives information about the number of Sylow subgroups.
The proofs of the theorems use nontrivial facts about group actions, in particular the action of on the coset space where has prime power order.
Let be a finite group. Let be a prime dividing Write with and
First Sylow Theorem. There is a subgroup of order is called a Sylow -subgroup.
Second Sylow Theorem. Any two Sylow -subgroups are conjugate: if and are Sylow -subgroups, there is an element such that
Third Sylow Theorem. Let be the number of Sylow -subgroups. Then
where is any Sylow -subgroup and denotes the normalizer of the largest subgroup of in which is normal.
Identify the Sylow subgroups of
The Sylow -subgroups have order There is a natural example of an order-8 subgroup in namely the dihedral group Labeling the vertices of a square the dihedral group is the set of symmetries of the square. It is generated by a -degree rotation, which is a -cycle, and a flip, which is a double transposition. There are four flips and four rotations (including the identity).
The dihedral group is not normal inside labeling the vertices differently leads to different copies. For instance, one copy is generated by and but another copy is generated by and and a third copy is generated by and These are all conjugate, as predicted by the second Sylow theorem; and meets the criteria of the third Sylow theorem: and
The Sylow -subgroups are generated by the -cycles, like There are such -cycles, hence four subgroups (each subgroup contains two of them). So This agrees with the third Sylow theorem, since and As for the third Sylow theorem predicts that this is a subgroup of order Let be one of the Sylow -subgroups, then is a copy of inside consisting of all the permutations that fix
Note that any conjugate of a Sylow -subgroup is a Sylow -subgroup. So if so that the Sylow -subgroup is unique, then it must be normal. This is a common source of applications.
Show that there is no simple group of order
Suppose is a simple group of order The third Sylow theorem for says that and So or If then the Sylow -subgroup is normal, which is impossible since is simple. So and there are Sylow -subgroups. They are distinct, so they intersect only in the identity element (the intersection is a subgroup, and by Lagrange's theorem its order is or so it must be ). Any Sylow -subgroup has order and consists of the identity and two elements of order So there are elements of order in
Now and so or As before, must be so there are Sylow -subgroups with elements each, consisting of the identity and four elements of order Any pair of subgroups intersects only in the identity, so there are a total of elements of order in But this count, coupled with the previous paragraph, gives at least elements in a group of order which is impossible. So there is no simple group of order
The Sylow theorems give extensive information about the structure of abelian groups (groups where the operation is commutative). Every subgroup of an abelian group is normal, so for all primes dividing So there is a unique Sylow -subgroup for every such prime
Next, there is a useful general lemma about abelian groups:
Let be subgroups of an abelian group and suppose Then is a subgroup of and it is isomorphic to the Cartesian product
Checking that is a group boils down to checking that it is closed under the group operation because is abelian. (Note that if is not abelian, as defined above is not necessarily a subgroup. Many references define to be the smallest subgroup containing all the elements for )
Note that if and are finite, with relatively prime orders, their intersection must be trivial by Lagrange's theorem. So the lemma applies to them.
Consider the effect of repeatedly applying the lemma to the Sylow -subgroups of an abelian group Suppose is the prime factorization of Let be the unique Sylow -subgroup. Then the lemma shows that So has order and the lemma applies to the groups and So
Proceeding in the same way gives the following result:
A finite abelian group is isomorphic to the Cartesian product of its Sylow subgroups.
This is a key component in the structure theorem for finite abelian groups.