Abelian and pronormal implies SCDIN

From Groupprops
Jump to: navigation, search
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)
View all subgroup property implications | View all subgroup property non-implications
Get more facts about abelian pronormal subgroup|Get more facts about SCDIN-subgroup


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


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