# Abelian Frattini subgroup implies centralizer is critical

## Contents

## Statement

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

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 -group, it is the unique smallest group such that the quotient is elementary Abelian.

In particular, for a -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 of a group is termed **critical** in if is a characteristic subgroup of and the following hold:

- ; In other words, is a Frattini-in-center group.
- : In other words, is a commutator-in-center subgroup of .
- : In other words, is a self-centralizing subgroup of .

## Related facts

- Centralizer of commutator subgroup has class at most two
- Centralizer of Frattini subgroup is Frattini-in-center

## Facts used

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

## Proof

**Given**: A finite -group with Abelian Frattini subgroup . .

**To prove**: is a critical subgroup of .

**Proof**: By facts (1) and (2), is characteristic in . Note also that since is Abelian, .

We now check each condition:

- : First, observe that , because is elementary Abelian. By definition of , we have . Thus, .
- : Indeed, .
- : We have , because . Thus, .