# Divisible abelian subgroup of abelian group contains no proper nontrivial verbal subgroup

From Groupprops

## Statement

Suppose is an abelian group, and is a subgroup of that is a Divisible abelian group (?). Then, if is a subgroup of that is a verbal subgroup of , then either is trivial or .

## Facts used

## Proof

**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 .