Finite index implies completely divisibility-closed

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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., subgroup of finite index) must also satisfy the second subgroup property (i.e., completely divisibility-closed subgroup)
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Statement

Suppose G is a group and H is a subgroup of finite index in G. Then, H is a completely divisibility-closed subgroup of G.

Facts used

  1. Poincare's theorem
  2. Subgroup of finite group implies completely divisibility-closed
  3. Divisibility-closedness satisfies inverse image condition

Proof

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