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

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 $S \setminus \operatorname{supp}(H)$.