Frattini subgroup

Symbol-free definition
The Frattini subgroup of a group is defined in the following equivalent ways:


 * 1) The intersection of all its defining ingredient::maximal subgroups (if there are no maximal subgroups, it equals the whole group)
 * 2) The set of all defining ingredient::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 defining ingredient::Frattini-embedded normal subgroup.

Definition with symbols
Let $$G$$ be a group. The Frattini subgroup of $$G$$, denoted $$\Phi(G)$$, is defined in the following equivalent ways:


 * 1) It is the intersection of all subgroups $$M \le G$$, where $$M$$ is maximal in $$G$$
 * 2) 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.

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.

Fixed-point operator
A group equals its own Frattini subgroup if and only if it has no maximal subgroups. For instance, the group of rational numbers, or more generally, the additive group of any field of characteristic zero has no maximal subgroups.

Free operator
A group whose Frattini subgroup is trivial is termed a Frattini-free group.

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:
 * ACIC-group:

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

Also related are:


 * Category:Terminology related to realization problems for Frattini subgroup
 * Category:Facts related to realization problems for Frattini subgroup

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).

Monotonicity
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, provided that normal subgroup satisfies the property that every proper subgroup is contained in a maximal subgroup. In particular, for any finite group, the Frattini subgroup of a normal subgroup is contained in the Frattini subgroup of the whole group.

Idempotence and iteration
The Frattini subgroup-defining function is not idempotent.

Associated constructions
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.

Textbook references

 * , Page 198-199
 * , Page 135
 * , Page 173, Section 5.1 (definition in paragraph, preceding Theorem 1.1)