Groups of prime order

Let be a Group.
We say that is a -group if for some and prime .

Theorem (centre)

If is a -group, then the Center is nontrivial.

Proof

Note that any Conjugacy Class has order or divisible by .
is the union of all classes.
The union of all other conjugacy classes is .
But size of all other classes is divisible by
so size of their union has to be divisible by
so has to be divisible by
so has to be divisible by
so (as nonempty with )

Corollary

If is a Simple -group then .

Corollary

Let be a -group of order .
Then contains a subgroup of order for every .

Proof

In the Simple Group Decomposition of we have:

where is simple.
Note that each is a -group,
hence so is .
But then .
The claim follows.

Lemma

For any Group , if is cyclic, then is Abelian.

Proof

Let be a generator of .
Then each element in can be written as for some and .
But now it is easy to check any two elements commute.

Corollary

Any Group of order for a prime is Abelian.