A characterization of normal subgroups via n-closed sets

TL;DR

This paper characterizes normal subgroups in groups using the concept of n-closed sets, showing that a subgroup is normal if and only if its left cosets are (n+1)-closed, linking algebraic structure to set closure properties.

Contribution

It introduces a novel characterization of normal subgroups via n-closed sets, extending the understanding of subgroup properties through closure concepts.

Findings

Normal subgroups correspond to (n+1)-closed left cosets.

Provides a new criterion for normality based on n-closed sets.

Links subgroup index to closure properties of cosets.

Abstract

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.