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 intersectionof all its maximal subgroups (if there are no maximal subgroups, it equals the whole group)
The set of all nongenerators, i.e., elements that can be removed from any generating set and still yield a generating set.

When the group has the property that every subgroup is contained in a maximal subgroup, the Frattini subgroup equals the unique largest Frattini-embedded normal subgroup.

Definition with symbols

Let $G$ be a group. The Frattini subgroup of $G$ , denoted $Φ (G)$ , is defined in the following equivalent ways:

It is the intersection of all subgroups $M \leq G$ , where $M$ is maximal in $G$
It is the set of all nongenerators $x$ , i.e., elements $x$ such that if $S \cup {x}$ is a generating set for $G$ , then so is $S$

When $G$ is a group in which every subgroup is contained in a maximal subgroup, then the Frattini subgroup is also the unique largest Frattini-embedded normal subgroup.

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. For full proof, refer: Nilpotent implies derived in Frattini

Group properties satisfied

In general, it is hard to find group properties satisfied by the Frattini subgroup of every group. However, for a group in which every subgroup is contained in a maximal subgroup, the Frattini subgroup usually satisfies some fairly strong restrictions. Most of these are restrictions that are satisfied by any Frattini-embedded normal subgroup. These include:

Inner-in-automorphism-Frattini group: For full proof, refer: Frattini-embedded normal implies inner-in-automorphism-Frattini
ACIC-group: For full proof, refer: Frattini-embedded normal implies ACIC

Since any finite ACIC-group is nilpotent, the Frattini subgroup of any finite group is nilpotent. 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.

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

Associated constructions

Associated quotient-defining function

The quotient-defining function associated with this subgroup-defining function is: Frattini quotient

Associated descending series

The associated descending series to this subgroup-defining function is: [[Frattini series]]

The Frattini series is the series obtained by iterating the Frattini subgroup operation, starting with the whole group. It gives aa descending series.

If this series terminates at the identity in finite length (which it will for a finite group, since the Frattini subgroup at each stage will be proper) then the length of the series is termed the Frattini length for the group.

References

Textbook references

Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347^{More info}, Page 198-199
A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613^{More info}, Page 135