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How come a transitive subgroup of $S_5$ must have order divisible by 5? I can see why any subgroup of $S_5$ with order divisible by 5 is transitive, but I do not see the reverse.

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The orbit-stabilizer theorem tells us that, if a finite group $G$ acts on a finite set $X$, then, for any $x$ in $X$, the order of $G$ is equal to the order of the stabilizer of $x$ times the size of the orbit of $x$. If the action is transitive, then every orbit is $X$ itself, so we have $|G|=|\mathrm{Stab}(x)||X|$, and so $|X|$ divides $|G|$. Applying this to the natural action of our subgroup on $\{1,2,3,4,5\}$ gives the result.

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