On Additive Combinatorics of Permutations of $\mathbb{Z}$_n

Let ${\mathbb{Z}}_n$ denote the ring of integers modulo $n$. In this paper we consider two extremal problems on permutations of ${\mathbb{Z}}_n$, namely, the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is again a permutation, and the maximum size of a collection of permutations such that the sum of any two distinct permutations in the collection is not a permutation. Let the sizes be denoted by $s(n)$ and $t(n)$ respectively. The case when $n$ is even is trivial in both the cases, with $s(n)=1$ and $t(n)=n!$. For $n$ odd, we prove $s(n)\geq (n\phi(n))/2^k$where$k$isthenumberofdistinctprimedivisorsof$n$\mathrm{.}When$n$isanoddprimeweprove$s(n)\leq \frac{e^2}{\pi} n ((n-1)/e)^\frac{n-1}{2}$\mathrm{.}Forthesecondproblem,weprove$2^{(n-1)/2}.(\frac{n-1}{2})!\leq t(n)\leq 2^k.(n-1)!/\phi(n)$when$n$ is odd.