Weakly normal subgroup of symmetric group must move more than half the elements

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Statement

Suppose S is a set, G = \operatorname{Sym}(S) is the Symmetric group (?) on S, and H is a nontrivial Weakly normal subgroup (?) of G. Define:

\operatorname{supp}(H) = \{ a \in S \mid \ \exists \ \sigma \in H, \sigma(a) \ne a \}.

Then, the cardinality of \operatorname{supp}(H) should be strictly greater than the cardinality of S \setminus \operatorname{supp}(H).

Note that since any Pronormal subgroup (?) and any Paranormal subgroup (?) is weakly normal, we get corresponding results for pronormal and paranormal subgroups. In fact, the proof also shows that the corresponding statement holds for Polynormal subgroup (?)s.