Central implies potentially verbal in finite

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

Suppose G is a finite group and H is a central subgroup of G. Then, there exists a finite group K containing G such that H is a Verbal subgroup (?) of K.

Related facts

Stronger facts and other corollaries of those stronger facts

Weaker facts

Other related facts

Facts used

  1. Central implies finite-pi-potentially verbal in finite

Proof

The result follows directly from fact (1), which is a stronger version of the same statement.