# 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. $\Phi(H) \le Z(H)$, viz the Frattini subgroup is contained inside the center (i.e., $H$ is a Frattini-in-center group).
2. $[G,H] \le Z(H)$ (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.