Extraspecial and critical implies whole group
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 ).
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.