Frattini-embedded normal-realizable group

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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
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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 $G$.

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