# Extraspecial and normal rank one implies quaternion group

## Statement

Let be a prime number and be a finite nontrivial -group. Suppose is an Extraspecial group (?): and this subgroup has order . Further, suppose that the normal rank of is one: every Abelian normal subgroup of is cyclic. Then, and is the quaternion group.

## Related facts

- Classification of finite p-groups of characteristic rank one
- Classification of finite p-groups of normal rank one
- Classification of finite p-groups of rank one

## Facts used

- Maximal among Abelian normal implies self-centralizing in nilpotent
- Classification of finite p-groups with cyclic normal self-centralizing subgroup

## Proof

### Proof outline

- We first show that : Every element of order is in the center.
- We next show that if , then the cyclic subgroup generated by has order , contains , and is a self-centralizing normal subgroup of .
- We finally use fact (2) to reduce to three possibilities for , and we eliminate two of them by inspection.

### Proof details

**Given**: An extraspecial -group of normal rank one.

**To prove**: and is isomorphic to the quaternion group.

**Proof**:

We first show that : Suppose such that has order . Then, commutes with , so the subgroup generated by and is elementary Abelian of order . Since it contains the commutator subgroup , it is also normal. Hence, we have an Abelian normal subgroup of that is not cyclic -- a contradiction. Thus, . On the other hand, clearly so .

Now, since is extraspecial, is elementary Abelian, so given any , . By the observations just made, has order . Let be the cyclic subgroup generated by . We claim that is a self-centralizing subgroup of . Clearly, is maximal among cyclic normal subgroups, since every element has order at most , and hence, by normal rank one, is maximal among Abelian normal subgroups. Thus, by fact (1), is a self-centralizing subgroup of .

Thus, has a self-centralizing cyclic normal subgroup of order . Fact (2) reduces us to three possibilities for . We easily see that two of these (the dihedral group of order eight and the non-Abelian group corresponding to an odd prime) do not satisfy the condition of normal rank one. This forces to be the quaternion group and .