A characterization of normal subgroups via -closed sets
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Let (G, *) be a semigroup, D ⊆ G, and n ≥ 2 be an integer. We say that (D, *) is an n-closed subset of G if a_1 * ... * a_n ∈ D for every a_1, ..., a_n ∈ 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 < ∞,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.
Abstract
Let (G, *) be a semigroup, D ⊆ G, and n ≥ 2 be an integer. We say that (D, *) is an n-closed subset of G if a_1 * ... * a_n ∈ D for every a_1, ..., a_n ∈ 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 < ∞,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.
참고문헌 (1)
- W. Hungerford, "Algebra" (1987), Springer-verlag