# Frattini subgroup is ACIC

This fact is related to the problem of: Frattini subgrouprealizationrelated to the following subgroup-defining function

Realization problems are usually about which groups can be realized as subgroups/quotients related to a subgroup-defining function.

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## Statement

### Verbal statement

The Frattini subgroup of a finite group (or more generally, of a group where every proper subgroup is contained in a maximal subgroup)is an ACIC-group.

### Symbolic statement

Let be a finite group (or more generally, a group where every subgroup is contained in a maximal subgroup), and be its Frattini subgroup. Then, is an ACIC-group: whenever is an automorph-conjugate subgroup of , is characteristic in .

## Definitions used

### Frattini subgroup

`Further information: Frattini subgroup`

The Frattini subgroup of a group is defined as the intersection of all its maximal subgroups.

### Group where every subgroup is contained in a maximal subgroup

For infinite groups, it could happen that there are no maximal subgroups, and it could also happen that not every proper subgroup is a maximal subgroup. The proof we give here does not work for such groups. We require our group to have the property that every proper subgroup is contained in a maximal subgroup.

This property is satisfied by finite groups, and more generally, by slender groups (the equivalent in group theory of Noetherian ring: a group where every subgroup is finitely generated).

### Automorph-conjugate subgroup

`Further information: automorph-conjugate subgroup`

A subgroup in a group is automorph-conjugate if for any automorphism of , and are conjugate.

Any characteristic subgroup is automorph-conjugate.

### ACIC-group

`Further information: ACIC-group`

An ACIC-group is a group in which every automorph-conjugate subgroup is characteristic. Equivalently, every automorph-conjugate subgroup is normal (The two ar eequivalent because normal automorph-conjugate subgroups are characteristic).

## Facts used

The proof is a generalized version of Frattini's argument.

## Generalizations

### To Frattini-embedded normal subgroups

`Further information: Frattini-embedded normal-realizable implies ACIC`
The result generalizes to the following result for arbitrary groups (with no finiteness assumptions): any Frattini-embedded normal subgroup of a group is ACIC. A Frattini-embedded normal subgroup is a normal subgroup whose product with any proper subgroup is proper.

## Proof

*Given*: a group with the property that every proper subgroup is contained in a maximal subgroup, and . is an automorph-conjugate subgroup of .

*To prove*: is a normal subgroup of

*Proof*: We will in fact show that is normal in . Normality in will then follow.

### Frattini's argument step

`Further information: Frattini's argument`

We first show that . This part is called the Frattini's argument. It only uses the hypothesis that is automorph-conjugate in and is normal in .

Suppose is any element. Then, since is normal in , conjugation by restricts to an automorphism of . Thus, and are related via an automorphism in .

Using the hypothesis that is automorph-conjugate inside , we see that there exists such that . Hence, the element lies in the normalizer . Rearranging, we see that any element of can be written as a product of an element of and an element of . So .

### Clinching step

This part uses the assumption we made about (that every proper subgroup is contained in a maximal subgroup) and the fact that is the Frattini subgroup of .

We assume that and obtain a contradiction. If , i.e. it is a proper subgroup, then by hypothesis it is contained in some maximal subgroup . By definition of Frattini subgrou, . So , yielding , a contradiction.

Thus, , hence is normal in .