Extending Potočnik and Šajna's conditions on the existence of vertex-transitive self-complementary k-hypergraphs

Let ` be a positive integer, k = 2 or k = 2 + 1, and let n be a positive integer with n ≡ 1 (mod 2). For a prime p, n(p) denotes the largest integer i such that p divides n. Potočnik and Šajna showed that if there exists a vertex-transitive self-complementary k-hypergraph of order n, then for every prime p we have p(p) ≡ 1 (mod 2). Here we extend their result to a larger class of integers k.