# Group Theory/Simple groups and Sylow's theorem

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**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:

- has a Sylow subgroup
- The action of by conjugation on the Sylow subgroups is transitive
- If is the number of Sylow -subgroups, then and
- 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).