# Special implies center is elementary abelian

## Contents

## Statement

In a special group, the center is an elementary Abelian group

## Proof

Let be the group and be the subgroup which is simultaneously the center, commutator subgroup and Frattini subgroup.

### 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.