# Frattini-embedded normal-realizable group

From Groupprops

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

This term 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.

View other terminology related to realization problems for Frattini subgroup OR View facts related to them

## Definition

A group is termed **Frattini-embedded normal-realizable** if there exists a group and an embedding of in such that is a Frattini-embedded normal subgroup of . In other words, is a normal subgroup and is a proper subgroup of for any proper subgroup of .

## Relation with other properties

### Stronger properties

### Weaker properties

- Inner-in-automorphism-Frattini group: This condition states that the inner automorphism group lies inside the Frattini subgroup of the automorphism group.
`For full proof, refer: Frattini-embedded normal-realizable implies inner-in-automorphism-Frattini` - ACIC-group:
`For full proof, refer: Frattini-embedded normal-realizable implies ACIC`

## Metaproperties

### Subgroups

This group property isnotsubgroup-closed, viz., we can have a group satisfying the property, with a subgroup not satisfying the property

### Quotients

This group property isnotquotient-closed, viz., we could have a group with the property and a quotient group of that group that doesnothave the property

### Characteristic subgroups

This group property is characteristic subgroup-closed: any characteristic subgroup of a group with the property, also has the property

View characteristic subgroup-closed group properties]]