Frattini-embedded normal-realizable group
BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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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
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
- 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
This group property is not subgroup-closed, viz., we can have a group satisfying the property, with a subgroup not satisfying the property
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
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]]