Characteristic subgroup of abelian group implies powering-invariant
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)
View all subgroup property implications | View all subgroup property non-implications
Get more facts about characteristic subgroup of abelian group|Get more facts about powering-invariant subgroup
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 .
- Characteristic not implies powering-invariant
- Characteristic subgroup of abelian group implies intermediately powering-invariant
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.