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. ◻