Is there any subgroup between $A_n$ and $A_{n+1}$

Question

Let $A_n$ be the alternating group of $n$ elements.

Is there any subgroup $H$ of $A_{n+1}$ such that $A_n \subsetneq H \subsetneq A_{n+1}$ for $n \geq 5$ ?

Actually, I'm working on the simplicity of $A_n$ for $n \geq 5$. And I find that it may help to consider this question.

Thanks for your help.

Answer 1

We view $A_n$ as the subgroup of $A_{n+1}$ as the subgroup that leaves $n+1$ fixed. Assume $A_n<H<A_{n+1}$. Then $h(n+1)\ne n+1$ for some $h\in H$. Let $g\in A_{n+1}$. Then $g\in H$:

• If $g(n+1)=n+1$, this is clear as $g\in A_n$.
• If $g(n+1)<n+1$, the fact that $A_n$ acts transitively on $\{1,\ldots,n\}$ allows us to find $a\in A_n$ with $g(n+1)=a(h(n+1))$. Then $b:=g^{-1}ah\in A_n$ and so $g=ahb^{-1}\in H$.