# Frattini-embedded normal-realizable implies every automorph-conjugate subgroup is characteristic

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 (i.e., Frattini-embedded normal-realizable group) must also satisfy the second group property (i.e., group in which every automorph-conjugate subgroup is characteristic)

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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

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.