  Let (G, *) be a semigroup, D subset of G, and n >= 2 be an integer. We say
that (D, *) is an n-closed subset of G if a_1* ... *a_n in D for every a_1,
..., a_n in D. Hence every closed set is a 2-closed set. The concept of
n-closed sets arise in so many natural examples. For example, let D be the set
of all odd integers, then (D, +) is a 3-closed subset of (Z, +) that is not a
2-closed subset of (Z, +). If K = {1, 4, 7, 10, ...}, then (K, +) is a 4-closed
subset of (Z, +) that is not an n-closed subset of (Z, +) for n = 2, 3. In this
paper, we show that if (H, *) is a subgroup of a group (G, *) such that [H: G]
= n < infty, then H is a normal subgroup of G if and only if every left coset
of $H$ is an (n+1)-closed subset of G.
