# Characteristic subgroup of abelian group implies powering-invariant

From Groupprops

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., characteristic subgroup of abelian group) must also satisfy the second subgroup property (i.e., powering-invariant subgroup)

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## Statement

Suppose is an abelian group and is a characteristic subgroup of . Then, is a powering-invariant subgroup of : for any prime number such that every element of has a unique root, every element of also has a unique root in .

## Related facts

- Characteristic not implies powering-invariant
- Characteristic subgroup of abelian group implies intermediately powering-invariant

## Facts used

## Proof

### Proof idea

The idea is to use Fact (1), and the powering, to show that the power map is an automorphism, hence so is its inverse (the root map), and hence, because the subgroup is characteristic, it is invariant under the map.