C-closed implies 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., 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 powering-invariant subgroup of G: for any prime number p such that G is p-powered, H is also p-powered.

Related facts

Applications

Facts used

  1. c-closed implies local powering-invariant
  2. Local powering-invariant implies powering-invariant

Proof

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