C-closed implies local 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., local powering-invariant subgroup)
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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 local powering-invariant subgroup of : for any natural number and any element such that the equation has a unique solution , we must have .
- C-closed implies powering-invariant
- Powering-invariance is centralizer-closed
- Center is local powering-invariant
- c-closed implies fixed-point subgroup of a subgroup of the automorphism group
- Fixed-point subgroup of a subgroup of the automorphism group implies local powering-invariant
The proof follows directly from Facts (1) and (2).
- Template:Paperlink-proved, Theorem 13.3, Page 229 (13th page in the paper)