C-closed 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., c-closed subgroup) 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 c-closed subgroup|Get more facts about powering-invariant subgroup
Suppose is a group and is a c-closed subgroup of , i.e., occurs as the centralizer of some subset (and hence also of some subgroup) of . Then, is a powering-invariant subgroup of : for any prime number such that is -powered, is also -powered.
The proof follows directly from Facts (1) and (2).