Weakly normal subgroup

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In group theory, a subgroup H of a group G is called weakly normal if the following holds: whenever the conjugate H^g is contained in the normalizer N_G(H) for some g ∈ G, then g itself lies in N_G(H). Here H^g denotes the conjugate of H by g (usually H^g = g^{-1}Hg), and N_G(H) = {x ∈ G : xHx^{-1} = H} is the normalizer of H in G. Intuition: if moving H by g keeps it inside its normalizer, then g already respects H as a symmetry of G. Fact: every pronormal subgroup is weakly normal.