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An easy proof that $S(n)$ does not embed into $A(n+1)$?
Here is a short proof that follows from Rotman's material: Automorphisms of $S_m$ are all inner unless $m=2 \text{ or } 6$. It is not hard to use this to show that for $m\neq 2, 6$ subgroups of $S_m$ ...
An easy proof that $S(n)$ does not embed into $A(n+1)$?
Using concepts that Rotman have introduced before that exercise, I think we could provide the following counterexample: if $S_2$ could be embedded in $A_3$, then in $A_3$ there would be a subgroup $H$ ...