Paranormal implies weakly normal
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., weakly normal subgroup)
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Statement with symbols
Suppose is a subgroup of such that is contranormal in for any . Then, if , we have .
Given: A group . A subgroup such that is contranormal in for any .
To prove: If , then .
Proof: Since , we have . In particular, is normal in . On the other hand, by paranormality, we have that the normal closure of in is . This forces , so .