A characterization of normal subgroups via \(n\)-closed sets.

zbMATH Open1252.20027arXiv1107.5058MaRDI QIDQ2918429

Ayman Badawi

Publication date: 6 October 2012

Published in: Advanced Studies in Contemporary Mathematics (Kyungshang) (Search for Journal in Brave)

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.

Full work available at URL: https://arxiv.org/abs/1107.5058

zbMATH Keywords

normal subgroupssubgroups of finite indexleft cosets\(n\)-closed subsets

Mathematics Subject Classification ID

Subgroup theorems; subgroup growth (20E07) Axiomatics and elementary properties of groups (20A05)

Related Items (1)

Closed normal subgroups

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