A characterization of normal subgroups via n-closed sets
Group Theory
2011-07-27 v1
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 is an (n+1)-closed subset of G.
Cite
@article{arxiv.1107.5058,
title = {A characterization of normal subgroups via n-closed sets},
author = {Ayman Badawi},
journal= {arXiv preprint arXiv:1107.5058},
year = {2011}
}