Abelian and pronormal implies SCDIN
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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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 is an abelian pronormal subgroup of a group . Then, is a SCDIN-subgroup of : given any two subsets of that are conjugate by in , they are conjugate by some in , and the action of coincides with the action of on .
The proof follows directly from facts (1) and (2).