Let ℤn denote the ring of integers modulo n. A permutation of ℤn is a sequence of n distinct elements of ℤn. Addition and subtraction of two permutations is defined element-wise. In this paper we consider two extremal problems on permutations of ℤn, 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 no sum of two distinct permutations in the collection is 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 (nφ(n))=2k ≥ s(n) ≥ n!-2-(n-1)=2 ((n-1)=2)! and 2(n-1)=2 - ( n-1 2 )! ≥ t(n) - 2k - (n - 1)!=φ(n), where k is the number of distinct prime divisors of n and φ is the Euler's totient function. © 2014 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.