Is it true that if a group $G$ operates on the set $S$ transitively, then $|S|$ must divide $|G|$? If yes, how to prove it? Thanks in advance!
2026-03-30 06:26:01.1774851961
Transitive group operation
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Yes, it is true, let $x\in S$, and $G_x$ the stabilizer of $x$, $|S|=|G|/|G_x|$. (Lagrange says that $|G_x|$ divides $|G|$).