Paranormal implies polynormal
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., paranormal subgroup) must also satisfy the second subgroup property (i.e., polynormal subgroup)
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For these definitions, denotes the conjugate of by , using the right-action convention (the action convention doesn't really matter). For subgroups , is the smallest subgroup of containing and invariant under the action of by conjugation.
Further information: Paranormal subgroup
Further information: Polynormal subgroup
Given: A paranormal subgroup of a group .
To prove: For any , is contranormal in .
Proof: Clearly, is generated by for all , which in turn means that it is generated by the subgroups . is contranormal in each of these by definition of paranormality, so by fact (1), is contranormal in .