Hall not implies pronormal
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., pronormal subgroup)
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By fact (1), construct a finite group and Hall subgroups of such that and are Hall of the same order but are not conjugate.
Let be a prime that does not divide the order of . Consider the wreath product of with the cyclic group of order acting regularly. This is a group given by:
Now consider the subgroups of given by:
- is a Hall subgroup of : is clearly a Hall subgroup of . is a Hall subgroup of because is a Hall subgroup of (fact (2)).
- is not pronormal in : Consider and its conjugate by a generator of the . We have . These are both subgroups in , hence if they are conjugate in the subgroup they are generate, they are conjugate in . However, if and are conjugate in , then the conjugating element acts coordinate-wise, so the first coordinate of (which is ) is conjugate to the first coordinate of (which is ) in . But this is contradictory to the assumption that and are not conjugate in .