TITLE: Is there a subset of $\Sigma_n$ s.t. each pair of elements is once in each pair of positions?
QUESTION [7 upvotes]: Is there a subset $A \subset \Sigma_n$ such that for each pair $(x, y)$ and each pair $(i, j)$, there is exactly one permutation $\sigma \in A$ such that $\sigma(i) = x$ and $\sigma(j) = y$? Remark that it implies that the cardinality of $A$ is $n * (n-1)$.
Is there a simple condition on $n$ such that it is possible?
If n is prime, it is easy, for example with the following method. For each pair $(x, k)$, consider the permutation $\sigma$ such that $\sigma(i) = x + (i - 1) * k$ (modulo $n$). For example, for $n = 5$, you get:

1 2 3 4 5 and its 5 circular permutations,
1 3 5 2 4 and its 5 circular permutations,
1 4 2 5 3 and its 5 circular permutations,
1 5 4 3 2 and its 5 circular permutations.
If n = 4, it is also possible. For example :
1 2 3 4
1 3 4 2
1 4 2 3
2 1 4 3
2 3 1 4
2 4 3 1
3 1 2 4
3 2 4 1
3 4 1 2
4 1 3 2
4 2 1 3
4 3 2 1
For $n = 6$, I suspect that it is not possible, but I am not sure...

REPLY [3 votes]: You are asking for a sharply $2$-transitive set of degree $n$. These objects are closely linked to projective planes of order $n$.
In particular, they exist if $n$ is a prime power, and no example is known with $n$ not a prime power. It also follows from the Bruck–Ryser Theorem that there is no example of degree $6$.
See for example 
http://tinyurl.com/go5el94
and 
http://www.mathematik.uni-wuerzburg.de/~mueller/Papers/sharplyTGPM.pdf