I think the following is sufficiently elementary: a transposition in $S_n$ is an element of order 2 commuting with at least $2(n-2)!$ elements of the group. But $A_{n+1}$ does not have such an element if $n$ is large enough. Indeed, if $\sigma\in A_{n+1}$ is of order 2, then it is a product of $k$ independent transpositions where $k$ is even and $2\le k\le(n+1)/2$. The number of elements of $A(n+1)$ A_{n+1}$commuting with such$\sigma$equals$2^{k-1}k!(n+1-2k)!$, and this is smaller than$2(n-2)!$provided that$n\ge 6$. 1 I think the following is sufficiently elementary: a transposition in$S_n$is an element of order 2 commuting with at least$2(n-2)!$elements of the group. But$A_{n+1}$does not have such an element if$n$is large enough. Indeed, if$\sigma\in A_{n+1}$is of order 2, then it is a product of$k$independent transpositions where$k$is even and$2\le k\le(n+1)/2$. The number of elements of$A(n+1)$commuting with such$\sigma$equals$2^{k-1}k!(n+1-2k)!$, and this is smaller than$2(n-2)!$provided that$n\ge 6\$.