Potentially verbal implies normal
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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Then, is a normal subgroup of .
Given: A group , a subgroup of , a group containing such that is a verbal subgroup of .
To prove: is normal in .
- By fact (1), is normal in .
- By fact (2), since is normal in , and is an intermediate subgroup of , then is normal in .