Let $G$ be a group acting on a set $A$. Both $|G|$ and $|A|$ are finite. If the action is transitive then $|A|$ divides $|G|$ (by orbit-stabiliser theorem). Is it true in general that $|A|$ divides $|G|$ ?
2026-03-30 20:41:00.1774903260
$G$ acts on $A$. Is it true that $|A|$ divides $|G|$
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No. Consider $G = \langle (12) \rangle$ and $A = \{ 1,2,3 \}$. Then $|A| = 3$ but $|G| = 2$.