Abelian and pronormal implies SCDIN

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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., abelian pronormal subgroup) must also satisfy the second subgroup property (i.e., SCDIN-subgroup)
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Statement

Verbal statement

An abelian pronormal subgroup (i.e., a Pronormal subgroup (?) that is also an abelian group) of a group is a SCDIN-subgroup: it is subset-conjugacy-determined in its normalizer relative to the whole group.

Statement with symbols

Suppose H is an abelian pronormal subgroup of a group G. Then, H is a SCDIN-subgroup of G: given any two subsets A,B of H that are conjugate by g in G, they are conjugate by some h in N_G(H), and the action of h coincides with the action of g on A.

Facts used

  1. Pronormal implies MWNSCDIN
  2. Abelian and MWNSCDIN implies SCDIN

Proof

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