P-Frattini-realizable implies not non-abelian cyclic-center

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This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property must also satisfy the second group property
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This fact 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 facts related to realization problems for Frattini subgroup OR View terminology related to them

Statement

Verbal statement

Any group of prime power order that is not Abelian, but has cyclic center, cannot be realized as the Frattini subgroup of a p-group.

References