2-subnormal not implies hypernormalized
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., 2-subnormal subgroup) need not satisfy the second subgroup property (i.e., hypernormalized subgroup)
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- Abnormal normalizer and 2-subnormal not implies normal: In fact, the same example used here works for that as well.
An example in the symmetric group on four letters
Let be the symmetric group on four letters and be the two-element subgroup generated by .
Then, is normal in the subgroup , which is normal in . So is 2-subnormal in .
On the other hand, the normalizer is a dihedral subgroup of order eight, which is a self-normalizing subgroup.