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This is an exercise in Dummit and Foote (4.6.3) I'm doing for revision: prove that for $n \geq 5$, $A_n$ is the only proper subgroup of $S_n$ such that $|S_n/G| < n$. ($A_n$ is the alternating group on $n$ elements and $S_n$ is the symmetric group on $n$ elements).

Not sure how to get started - given the condition that $n \geq 5$ it seems that the fact that $A_n$ is simple should be useful.

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I assume that you did the exercise above it that asks you to find all the normal subgroups of $S_n$ ($n\geq 5)$. You should get that the only normal subgroups are the trivial, the alternating, and the whole group.

Now to tackle your problem, let $K$ be a proper subgroup of $S_n$ such that $|S_n/K|=k