Finite-Frattini-realizable group

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This article defines a property that can be evaluated for finite groups (and hence, a particular kind of group property)
View other properties of finite groups OR View all group properties

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

Symbol-free definition

A finite group is termed finite-Frattini-realizable if it can be realized as the Frattini subgroup of some finite group.

Definition with symbols

A finite group $G$ is termed finite-Frattini-realizable if there exists a finite group $K$ such that $\Phi (K)\cong G$ where $\Phi (K)$ denotes the Frattini subgroup of $K$.

Relation with other properties

Weaker properties

• Finite ACIC-group: For full proof, refer: Frattini subgroup is ACIC
• Finite nilpotent group
• Frattini-realizable group: A group that can be realized as the Frattini subgroup of a not necessarily finite group
• Finite-(Frattini-embedded normal)-realizable group