Free implies every subgroup is descendant

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This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., free group) must also satisfy the second group property (i.e., group in which every subgroup is descendant)
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Statement

In a free group, every subgroup is a descendant subgroup.

Facts used

  1. Free implies residually nilpotent
  2. Residually nilpotent implies hypocentral
  3. Hypocentral implies every subgroup is descendant

Proof

The proof follows by combining Facts (1)-(3).