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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
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This term 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 terminology related to realization problems for Frattini subgroup OR View facts related to them
Definition
A group N is termed Frattini-embedded normal-realizable if there exists a group G and an embedding of N in G such that N is a Frattini-embedded normal subgroup of G. In other words, N is a normal subgroup and NH is a proper subgroup of G for any proper subgroup H of G.
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 is not subgroup-closed, viz., we can have a group satisfying the property, with a subgroup not satisfying the property
Quotients
This group property is not quotient-closed, viz., we could have a group with the property and a quotient group of that group that does not have 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 other characteristic subgroup-closed group properties

