C-closed implies local powering-invariant

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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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Statement

Suppose G is a group and H is a c-closed subgroup of G, i.e., H occurs as the centralizer of some subset (and hence also of some subgroup) of G. Then, H is a local powering-invariant subgroup of G: for any natural number n and any element h \in H such that the equation x^n = h has a unique solution x \in G, we must have x \in H.

Related facts

Applications

Facts used

  1. c-closed implies fixed-point subgroup of a subgroup of the automorphism group
  2. Fixed-point subgroup of a subgroup of the automorphism group implies local powering-invariant

Proof

The proof follows directly from Facts (1) and (2).

References

Journal references