# Characteristic subgroup of abelian group not implies local powering-invariant

From Groupprops

This article gives the statement and possibly, proof, of a non-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) neednotsatisfy the second subgroup property (i.e., local powering-invariant subgroup)

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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a abelian group. That is, it states that in a abelian group, every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) neednotsatisfy the second subgroup property (i.e., local powering-invariant subgroup)

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## Statement

It is possible to have an abelian group and a characteristic subgroup of that is not a local powering-invariant subgroup of : in other words, it is possible to have and are such that there is a unique satisfying , but .

## Related facts

### Similar facts

### Opposite facts

- Verbal subgroup of abelian group implies divisibility-closed
- Characteristic subgroup of abelian group implies powering-invariant

## Proof

Set and as the subgroup .

- is charcateristic in : In fact, is a verbal subgroup of .
- is not local powering-invariant in : The element has a unique square root (corresponding to ) in , but this square root is not in .