Hall not implies WNSCDIN

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

• Sylow implies WNSCDIN: This is a combination of the facts Sylow implies pronormal and pronormal implies WNSCDIN.
• Hall not implies procharacteristic
• Hall not implies pronormal

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]

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