Prove the group transitivity of alternating group $A_n \quad n>2$?

Question

Does it not suffice to point out that

$$(i, k)(i, j)\in A_n$$

The element at location $i$ is mapped to the element at location $j$ and and the element at location $j$ is mapped to some third location $k$.

Does this not prove transitivity if $n>2$?

Answer 1

Yes: if $n>2$ then we can pick $i,j$ arbitrary and any $k\neq i,j$ and then $(ik)(ij):i\mapsto j$ and we know that $(ik)(ij)\in A_n$. (It doesn't matter where it sends $k$.)

In fact $A_n$ acts $(n-2)$-transitively. Given any two lists of all but $2$ numbers from $\{1,\cdots,n\}$, there are exactly two permutations that send the $i$th number in the first list to the $i$th number in the second list for each $i=1,\cdots,n-2$. Such a permutation's outputs are determined on numbers in the first list - the only choices that get to be made in the construction is where we map the two numbers not in the first list to the two numbers not in the second list. If $\alpha$ and $\beta$ are the two permutations and $u,v$ are the two numbers not in the second list, then $\alpha=(uv)\beta$, so exactly one of the two permutations is in $A_n$.