Powering-invariant not implies local powering-invariant
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) need not satisfy the second subgroup property (i.e., local powering-invariant)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about powering-invariant subgroup|Get more facts about local powering-invariant
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property powering-invariant subgroup but not local powering-invariant|View examples of subgroups satisfying property powering-invariant subgroup and local powering-invariant
- 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 .
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 .