Abelian Frattini subgroup implies centralizer is critical
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 (?).
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.
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 .
- Centralizer of commutator subgroup has class at most two
- Centralizer of Frattini subgroup is Frattini-in-center
- Frattini subgroup is characteristic
- Characteristicity is centralizer-closed: The centralizer of a characteristic subgroup is characteristic.
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, .