Paranormal not implies pronormal

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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., paranormal subgroup) need not satisfy the second subgroup property (i.e., pronormal subgroup)
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Statement

A paranormal subgroup of a group need not be pronormal.

Definitions used

Pronormal subgroup

Further information: Pronormal subgroup

A subgroup H of a group G is termed pronormal in G if for any g \in G there exists x \in \langle H, H^g \rangle such that H^x = H^g.

Paranormal subgroup

Further information: Paranormal subgroup

A subgroup H of a group G is termed paranormal in G if for any g \in G, H is a contranormal subgroup of \langle H, H^g \rangle: in other words, the normal closure of H in \langle H, H^g \rangle is the whole group \langle H, H^g.

Proof

Example of the symmetric group on six elements

Further information: symmetric group:S6

Let G be the symmetric group on the set \{ 1,2,3,4,5,6 \}. 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:

  1. H_1 := \langle (4,5,6) , (2,3)(5,6) \rangle:
    • This is paranormal
    • This is not pronormal
  2. H_2 := \langle (1,2,3)(4,5,6), (2,3)(5,6) \rangle:
    • This is paranormal
    • This is not pronormal
  3. H_3 := \langle (1,2,3), (2,3), (4,5) \rangle. In other words, H_3 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
  4. H_4 := \langle (1,2)(3,4)(5,6), (1,3,5)(2,4,6), (3,5)(4,6) \rangle
    • This is paranormal
    • This is not pronormal