Powering-invariant not implies divisibility-closed
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., divisibility-closed subgroup)
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- is a powering-invariant subgroup of : If is a natural number such that every element of has a unique root, then every element of has a unique root within .
- is not a divisibility-closed subgroup of : There exists a natural number such that every element of has a root (not necessarily unique) but not every element of has a root within .
The key fact is that any finite subgroup of a group must be powering-invariant, but it need not be divisibility-closed. We will construct an example where the subgroup is finite.
For any prime number :
- Let be the -quasicyclic group.
- Let be the subgroup comprising the elements of order 1 or .
- , being finite, is powering-invariant (in fact, both and are powered over precisely the set of primes other than ).
- However, is not divisibility-closed: is -divisible, but is not.