Let ℤ_n denote the ring of integers modulo n. A permutation of ℤ_n is a sequence of n distinct elements of ℤ_n. Addition and subtraction of two permutations is defined element-wise. In this paper we consider two extremal problems on permutations of 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 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))/2^k≤s(n)≤n!· 2^-(n-1)/2((n-1)/2)! and 2^(n-1)/2·(n-1 / 2)!≤t(n)≤ 2^k·(n-1)!/φ(n), where k is the number of distinct prime divisors of n and φ is the Euler's totient function.