Extraspecial and critical implies whole group

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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.