Potentially verbal implies normal

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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., potentially verbal subgroup) must also satisfy the second subgroup property (i.e., normal subgroup)
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Statement

Suppose G is a group and H is a potentially verbal subgroup of G, i.e., there exists a group K containing G such that H is a verbal subgroup of K.

Then, H is a normal subgroup of G.

Facts used

  1. Verbal implies normal
  2. Normality satisfies intermediate subgroup condition

Proof

Given: A group G, a subgroup H of G, a group K containing G such that H is a verbal subgroup of K.

To prove: H is normal in G.

Proof:

  1. By fact (1), H is normal in K.
  2. By fact (2), since H is normal in K, and G is an intermediate subgroup of K, then H is normal in G.