Centers and centralizers

Definition: Centralizer, center

Let be a group and be any subset. The centralizer of is and the center of is

Remark

Every centralizer is a subgroup since if then . Also, is the kernel of where , since if and only if for every , if and only if for every . Finally, since it is a kernel, and .

Theorem

A group is abelian if and only if is cyclic.

Proof:
If is abelian, so is the trivial group and therefore cyclic.

Now assume is cyclic, so for some . Take any . We get and , meaning and for some , which in turn means

Definition: Representative, system of representatives

For any orbit , an element is called a representative of .

A system of representatives for a family of orbits is a family of representatives where is a representative of for every .

Theorem: The Class Equation

Consider the conjugation action of a finite group and let be a system of representatives of the orbits contained in (where '' denotes setminus). Then Proof:
The point of the theorem is that can be partitioned into its conjugacy classes, which are the equivalence classes with conjugacy as equivalence relation. We know each element in has only itself as conjugacy class, since every conjugation does nothing. Hence instead of writing out for every element in we combine them into just . So left to sum is the sizes of all conjugacy classes larger than one. We see that since our group action is conjugation, the conjugacy classes are precisely the orbits. Also, we see that is just the Stabilizer Stabilizer of , meaning is indeed the size of the orbit of , by the Orbit stabilizer theorem Orbit stabilizer theorem.

Corollary

If where is prime, then is abelian.

Proof:
We know is a normal subgroup, meaning its order divides , meaning it's either 1, , or .

If , then , meaning is abelian.
If , then , meaning is cyclic by Theorem Theorem, meaning is abelian. Note that since is abelian, , so this case doesn't even happen.
If , the Class equation Class equation gives that where we know each is either , or , and since they are all greater than (since only the orbits of elements of have size ), it's either or . This means the right hand side is divisible by , but clearly the left hand side isn't, meaning .