Abelian Frattini subgroup implies centralizer is critical

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Statement

Suppose P is a group of prime power order such that the Frattini subgroup (?) \Phi(P) of P is an Abelian group (?). Then, the centralizer of the Frattini subgroup, i.e., the group C_P(\Phi(P)), is a Critical subgroup (?) of P.

Note that the critical subgroup we obtain this way is of a special kind: it is a C-closed critical subgroup (?).

Definitions used

Frattini subgroup

Further information: Frattini subgroup

The Frattini subgroup of a group is defined as the intersection of all its maximal subgroups. For a p-group, it is the unique smallest group such that the quotient is elementary Abelian.

In particular, for a p-group, the Frattini subgroup contains the commutator subgroup, and it also contains the Frattini subgroup of any intermediate subgroup.

Critical subgroup

Further information: Critical subgroup

A subgroup C of a group P is termed critical in P if C is a characteristic subgroup of P and the following hold:

  1. \Phi(C) \le Z(C); In other words, C is a Frattini-in-center group.
  2. [P,C] \le Z(C): In other words, C is a commutator-in-center subgroup of P.
  3. C_P(C) \le C: In other words, C is a self-centralizing subgroup of P.

Related facts

Facts used

  1. Frattini subgroup is characteristic
  2. Characteristicity is centralizer-closed: The centralizer of a characteristic subgroup is characteristic.

Proof

Given: A finite p-group P with Abelian Frattini subgroup \Phi(P). C = C_P(\Phi(P)).

To prove: C is a critical subgroup of P.

Proof: By facts (1) and (2), C is characteristic in P. Note also that since \Phi(P) is Abelian, \Phi(P) \le C.

We now check each condition:

  1. \Phi(C) \le Z(C): First, observe that \Phi(C) \le \Phi(P), because C/\Phi(P) is elementary Abelian. By definition of C, we have \Phi(P) \le Z(C). Thus, \Phi(C) \le Z(C).
  2. [P,C] \le Z(C): Indeed, [P,C] \le [P,P] \le \Phi(P) \le Z(C).
  3. C_P(C) \le C: We have C_P(C) \le C_P(\Phi(P)), because \Phi(P) \le C. Thus, C_P(C) \le C.