Hall not implies WNSCDIN

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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., Hall subgroup) need not satisfy the second subgroup property (i.e., WNSCDIN-subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
Get more facts about Hall subgroup|Get more facts about WNSCDIN-subgroup
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property Hall subgroup but not WNSCDIN-subgroup|View examples of subgroups satisfying property Hall subgroup and WNSCDIN-subgroup

Statement

There exists a finite group G and a Hall subgroup H of G such that H is not a WNSCDIN-subgroup of G. In other words, there exist subsets A,B of H such that A and B are conjugate in G but not in N_G(H).

Related facts

Proof

(The same general example used to show that Hall subgroups need not be pronormal works here). PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]