Frattini subgroup

This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions

Definition

Symbol-free definition

The Frattini subgroup of a group is defined in the following equivalent ways:

• The intersection of all its maximal subgroups
• The set of all nongenerators

Relation with other subgroup-defining functions

Larger subgroup-defining functions

• Fitting subgroup always contains the Frattini subgroup (for a finite group) This follows from the fact that the Frattini subgroup is nilpotent.

Smaller subgroup-defining functions

• Commutator subgroup when the whole group is nilpotent

Group properties satisfied

The Frattini subgroup of any finite group is a nilpotent group. This follows from Frattini's argument. For full proof, refer: Frattini subgroup is nilpotent

Subgroup properties satisfied

The Frattini subgroup of any group is a characteristic subgroup.

For a finite group, the Frattini subgroup is always a proper subgroup (because there exist maximal subgroups).

Subgroup-defining function properties

Monotonicity

This subgroup-defining function is normal-monotone, viz applying the subgroup-defining function to a normal subgroup gives a smaller subgroup than applying it to the whole group

The Frattini subgroup-defining function is not monotone. However, the Frattini subgroup of any normal subgroup is contained in the Frattini subgroup of the whole group. Hence, it is a normal-monotone subgroup-defining function.

For full proof, refer: Frattini subgroup is normal-monotone

Idempotence and iteration

The Frattini subgroup-defining function is not idempotent. Iteration of the Frattini subgroup-defining function gives the Frattini series of a group. For any finite nontrivial group, the Frattini subgroup is always proper and hence the Frattini series terminates after a finite length at the trivial subgroup.

Quotient-idempotence

This subgroup-defining function is quotient-idempotent: taking the quotient of any group by the subgroup, gives a group where the subgroup-defining function yields the trivial subgroup
View a complete list of such subgroup-defining functions