Extraspecial implies Camina

Statement
Any extraspecial group is a Camina group. In other words, if $$P$$ is a group of prime power order (or more generally, a possibly infinite p-group) such that the center, Frattini subgroup, and commutator subgroup of $$P$$ all coincide and are cyclic of prime order, then $$P$$ is a Camina group.

Proof
Given: A $$p$$-group $$P$$ such that $$Z(P) = \Phi(P) = [P,P]$$ is cyclic of prime order.

To prove: Every coset of $$[P,P]$$ other than $$[P,P]$$ itself is a conjugacy class.

Proof: Since $$[P,P]$$ is of order $$p$$, so is every coset of it. Each coset of $$[P,P]$$ is a union of conjugacy classes (since nay two conjugate elements are in the same coset of $$[P,P]$$. Thus, the size of each conjugacy class is at most $$p$$. On the other hand, the conjugacy class of $$g \in P$$ is in bijection with the coset space of $$C_P(g)$$. Since $$P$$ is a $$p$$-group, any subgroup of finite index has index a power of $$p$$, so the size of every conjugacy class is a power of $$p$$. Thus, the only possible sizes of conjugacy classes are $$1$$ and $$p$$.

However, since $$Z(P) = [P,P]$$, no element outside $$[P,P]$$ has a conjugacy class of size one, forcing all conjugacy classes outside to have size $$p$$. Hence, for any element outside $$[P,P]$$, its conjugacy class equals its $$[P,P]$$-coset.