Group Theory/Simple groups and Sylow's theorem

Definition (Sylow p-subgroup):

Let be a group and let be a prime number such that . Then a Sylow -subgroup of is a subgroup such that , where is maximal such that .

Theorem (Cauchy's theorem):

Let be a group whose order is divisible by a prime number . Then contains an element of order .

Proof: acts on itself via conjugation. Let be a system of representatives of cojugacy classes. The class equation yields


Either, there exists such that is both not and not divisible by , in which case we may conclude by induction on the group order, noting that divides and , or for all the number is either or divisible by ; but in this case, by taking the class equation , we obtain that is nontrivial and moreover that its order is divisible by . Hence, it suffices to consider the case where is an abelian group. Take then any element . If has order divisible by , raising to a sufficiently high power will produce an element of order . Otherwise, the order of is divisible by , and by induction we find an element whose order is divisible by . Then the order of will also be divisible by , because otherwise, passing to the quotient, for some not divisible by .

Theorem (Sylow's theorem):

Let be a finite group, such that with . Then the following hold:

  1. has a Sylow subgroup
  2. The action of by conjugation on the Sylow subgroups is transitive
  3. If is the number of Sylow -subgroups, then and
  4. Every -group of is contained within some Sylow -group of

Definition (simple group):

A group is a simple group if and are the only normal subgroups of (where denotes the identity).