Paranormal implies weakly normal

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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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Verbal statement

A paranormal subgroup of a group must be a weakly normal subgroup.

Statement with symbols

Suppose H is a subgroup of G such that H is contranormal in \langle H, H^g \rangle for any g \in G. Then, if H^g \le N_G(H), we have H^g \le H.

Related facts


Given: A group G. A subgroup H such that H is contranormal in \langle H, H^g \rangle for any g \in G.

To prove: If H^g \le N_G(H), then H^g \le H.

Proof: Since H^g \le N_G(H), we have \langle H, H^g \rangle \le N_G(H). In particular, H is normal in \langle H, H^g \rangle. On the other hand, by paranormality, we have that the normal closure of H in \langle H, H^g \rangle is \langle H, H^g \rangle. This forces H = \langle H, H^g \rangle, so H^g \le H.