Paranormal not implies pronormal
From Groupprops
This article gives the statement and possibly, proof, of a nonimplication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., paranormal subgroup) need not satisfy the second subgroup property (i.e., pronormal subgroup)
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Contents
Statement
A paranormal subgroup of a group need not be pronormal.
Definitions used
Pronormal subgroup
Further information: Pronormal subgroup
A subgroup of a group is termed pronormal in if for any there exists such that .
Paranormal subgroup
Further information: Paranormal subgroup
A subgroup of a group is termed paranormal in if for any , is a contranormal subgroup of : in other words, the normal closure of in is the whole group .
Proof
Example of the symmetric group on six elements
Further information: symmetric group:S6
Let be the symmetric group on the set . Then there are in fact four different conjugacy classes of subgroups that are paranormal but not pronormal. We list these examples (by providing a representative subgroup for each) and explain why each one works:
 :
 This is paranormal
 This is not pronormal
 :
 This is paranormal
 This is not pronormal
 . In other words, is the product of the symmetric group on the first three elements and the symmetric group on the fourth and fifth elements.
 This is paranormal
 This is not pronormal

 This is paranormal
 This is not pronormal