Special implies center is elementary abelian
The Frattini part
The fact that is the Frattini subgroup is used up in the observation that is elementary Abelian.
The commutator subgroup part
The fact that is the commutator subgroup is used in the observation that commutators viz elements of the form where , generate .
The center part
Since is the center, the map given by descends to a map from to . As observed earlier, is elementary Abelian, so for any .
Now since the center is the same as the commutator, the commutator map is a bihomomorphism, and we have:
Since by the above observation. Thus every commutator has order or .
Putting the pieces together
We have the following facts:
- is Abelian
- is generated by elements having order
Putting the pieces together, we see that every element in has order so is elementary Abelian.