Divisible abelian subgroup of abelian group contains no proper nontrivial verbal subgroup
Given: An abelian group , a subgroup of that is divisible, and a subgroup of that is verbal in .
To prove: or is trivial.
Proof: By fact (1), there exists an integer such that is the set of powers in . If , is trivial. If, on the other hand, is nonzero, then every element of is a power, so . Since by assumption, we get .