# Powering-invariant 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., powering-invariant subgroup) neednotsatisfy the second subgroup property (i.e., local powering-invariant)

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## Statement

It is possible to have a group and a powering-invariant subgroup of that is not local powering-invariant. In other words, the following are true:

- is powering-invariant in : For any prime number , if is powered over , is also powered over .
- is not local powering-invariant in : There exists a natural number and an element such that there is a unique satisfying , but .

## Proof

Set and as the subgroup .

- is powering-invariant in : is not powered over any prime, so is is powering-invariant in for vacuous reasons.
- is not local powering-invariant in : The element has a unique square root (corresponding to ) in , but this square root is not in .