Central implies finite-pi-potentially verbal in finite

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a Finite group (?), every subgroup satisfying the first subgroup property (i.e., Central subgroup (?)) must also satisfy the second subgroup property (i.e., Finite-pi-potentally verbal subgroup (?)). In other words, every central subgroup of finite group is a finite-pi-potentally verbal subgroup of finite group.
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Statement

Suppose G is a finite group and H is a central subgroup of G, i.e., H is contained inside the center of G. Then, there exists a finite group K containing G such that every prime divisor of the order of K also divides the order of G, and H is a Verbal subgroup (?) of K.

Related facts

Weaker facts

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