# Extraspecial and critical implies whole group

From Groupprops

## Statement

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

## Definitions used

### Critical subgroup

`Further information: critical subgroup`

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

- , viz the Frattini subgroup is contained inside the center (i.e., is a Frattini-in-center group).
- (i.e., is a commutator-in-center subgroup of ).
- (i.e., is a self-centralizing subgroup of ).

## Facts used

## Proof

**Given**: A finite -group , a critical subgroup that is also extraspecial.

**To prove**: .

**Proof**: By point (2) of the definition of critical subgroup, is a commutator-in-center subgroup of . Combining this with fact (1) yields that is a central factor of . Thus, .

Point (3) of the definition of critical subgroup says that , so , so , completing the proof.