The Group Properties Wiki (pre-alpha)

TIP: Get tips on handling doubts and solving riders

ABOUT US: Read our purpose statement and learn what makes us special

ALSO CHECK OUT: Diffgeom: The Differential Geometry Wiki

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property must also satisfy the second group property
View all group property implications | View all group property non-implications |Get help on looking up group property implications/non-implications
|

This fact is related to the problem of realization related to the following subgroup-defining function: Frattini subgroup
Realization problems are usually about which groups can be realized as subgroups/quotients related to a subgroup-defining function.
View other facts related to realization problems for Frattini subgroup OR View terminology related to them

Contents 1 Statement 1.1 Verbal statement 2 Related facts 3 Intermediate properties 4 Proof

Statement

Verbal statement

Any Frattini-embedded normal-realizable group (i.e. any group that occurs as a Frattini-embedded normal subgroup of some group) must be an ACIC-group.

Related facts

A special case of this is that for a finite group, the Frattini subgroup is ACIC.

Intermediate properties

• Inner-in-automorphism-Frattini group: This property lies in between the property of being Frattini-embedded normal-realizable, and the property of being an ACIC-group.

Proof