Pronormal not implies NE
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., pronormal subgroup) need not satisfy the second subgroup property (i.e., NE-subgroup)
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Examples of Sylow subgroups
The proof that pronormal subgroups need not be NE follows from facts (1) and (2). Further, any example of a Sylow subgroup that is not NE gives an example of a pronormal subgroup that is not NE. Two examples of situations where Sylow subgroups are not NE are the -Sylow subgroup and the -Sylow subgroup in the alternating group of degree five.
Further information: Sylow not implies NE
Example of the symmetric group
Further information: symmetric group:S4
Let be the symmetric group on the set , and be the four-element subgroup . Then, is a pronormal subgroup, because the subgroup generated by and any other conjugate of it is the whole group. On the other hand, we have and is a dihedral group of order eight, so is a dihedral group of order eight, which is strictly bigger than .