If we have a group $G$ acting faithfully on a set of size n, could we say the size of the group is at most $n!$? If so, why?
2026-04-21 20:50:54.1776804654
Group order from the orbit size in a faithful action.
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If $G$ acts on a set $X$ via an action $G \times X \to X, (g,x) \mapsto gx$, then the map $G \to \text{Sym}(X), g \to (x \mapsto gx)$ is a homomorphism of groups. It is not difficult to show that the action is faithful if and only if this homomorphism is injective. Finally, recall that for finite $X$ we have $|\text{Sym}(X)| = |X|!$.