Alternating group $A_n$ where $n\geq 5$

Question

I tried so much to prove the following fact about the alternating groups $A_n$, $n\geq 5$ but I couldn't prove it. Any answer or hint will be appreciate;

Any maximal subgroup of the mentioned groups has size greater than n.

Many thanks!

Answer 1

Consider a maximal subgroup $H$ of $\mathrm{Alt}(\{1,\dots,n\})$. The orbits $X_1,\dots,X_k$ of $H$ form a partition of $\{1,\dots,n\}$. If there is only one orbit ($H$ is transitive) then $|H|\geq |X_1|=n$. Otherwise $H$ is contained in $H'=\mathrm{Alt}(\{1,\dots,n\})\cap(\mathrm{Sym}(X_1)\times\mathrm{Sym}(X_2\cup\dots\cup X_k))$, and by maximality of $H$ we have $H=H'$. So $|H|=|H'|=|X_1|! (n-|X_1|)!/2$, which is at least $n$ for $n\geq 5$.