Extraspecial and critical implies whole group

Statement

Suppose $G$ is a Group of prime power order (?), and $H$ is a Critical subgroup (?) of $G$ that is also an Extraspecial group (?). Then, $H=G$.

Definitions used

Critical subgroup

Further information: critical subgroup

A characteristic subgroup $H$ of a finite $p$-group $G$ is termed critical if it satisfies the following conditions:

1. $\Phi (H)\leq Z(H)$, viz the Frattini subgroup is contained inside the center (i.e., $H$ is a Frattini-in-center group).
2. $[G,H]\leq Z(H)$ (i.e., $H$ is a commutator-in-center subgroup of $G$).
3. $C_{G}(H)=Z(H)$ (i.e., $H$ is a self-centralizing subgroup of $G$).

Facts used

1. Extraspecial commutator-in-center subgroup is central factor

Proof

Given: A finite $p$-group $G$, a critical subgroup $H$ that is also extraspecial.

To prove: $H=G$.

Proof: By point (2) of the definition of critical subgroup, $H$ is a commutator-in-center subgroup of $G$. Combining this with fact (1) yields that $H$ is a central factor of $G$. Thus, $HC_{G}(H)=G$.

Point (3) of the definition of critical subgroup says that $C_{G}(H)=Z(H)$, so $HZ(H)=G$, so $H=G$, completing the proof.