Group actions and Sylow theorems

If a positive integer divides the order of a finite group , does have a subgroup of order ? The answer is true for finite abelian groups, but it is not true for arbitrary groups. Sylow theorems discuss this situation when is a prime power.

Before discussing Sylow theorems, we first discuss group actions.

Let is a group and , the action of group on the set where is the product on is called a left translation. The action of on where is called conjugation by and the element is said to be a conjugate of . If is any subgroup of and , . Thus acts on the set of all subgroups of by conjugation . The group is said to be conjugate to .

The equivalence classes are called the orbits of on , denoted by for . The group is called the stabilizer of . If acts on itself by conjugation, the orbits are called conjugacy classes. If a subgroup acts on by conjugation, is called the centralizer of in and is denoted . is simply called the centralizer of . If acts by conjugation on the set of subgroups of , the subgroup of fixing , is called the normalizer of and is denoted . The group is simply the normalizer of . Every subgroup is normal in and is normal iff .

Proof. Let . Since it follows that is a well-defined bijection of the set of cosets of in onto . Hence . ◻

Proof. (i) and (iii) follow from the previous theorem and Lagrange’s theorem. Since conjugacy is an equivalence relation, (ii) follows from (i). ◻

Proof. If , define by . Since , is surjective. Similarly, implies whence is injective. Since , the map given by is a homomorphism. ◻

Proof. Let act on itself by left translation and obtain . If , then . In particular, whence and is a monomorphism. Note if , . ◻

If is a group, , the set of all automorphisms of is a group under composition.

Proof. (i) If acts on itself by conjugation, given by is a bijection. is also a homomorphism and hence an automorphism. (ii) Let act on itself by conjugation. The homomorphism has image contained in . Clearly whence . ◻

The automorphism is called the inner automorphism induced by . is called the center of . An element iff the conjugacy class of consists of alone. Thus if , then . Thus if is finite, then where are distinct conjugacy classes of and each . The above equation is called the class equation.

Proof. The induced homomorphism is given by where and . If and . In particular, implying . ◻

Proof. Apply the proposition. The kernel of the induced homomorphism is a normal subgroup of contained in and thus must be . Hence is a monomorphism. ◻