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Ars Combinatoria

  • Research article

Nilpotent Covers of Dihedral Groups

Kimeu Arphaxad Ngwava1, Nick Gill 2
1P.O.BOX 116–90100, Machakos,Kenya; Moi University P.O.B0X 3900–30100, Eldoret, Kenya
2Department of Mathematics, University of South Wales, Treforest, CF37 1DL, U.K.
  • Received: 15/03/2020
  • Revised: 28/05/2020
  • Published: 30/09/2024

Abstract

Let \(G\) be a group, and let \(c\in\mathbb{Z}^+\cup\{\infty\}\). We let \(\sigma_c(G)\) be the maximal size of a subset \(X\) of \(G\) such that, for any distinct \(x_1,x_2\in X\), the group \(\langle x_1,x_2\rangle\) is not \(c\)-nilpotent; similarly we let \(\Sigma_c(G)\) be the smallest number of \(c\)-nilpotent subgroups of \(G\) whose union is equal to \(G\). In this note we study \(D_{2k}\), the dihedral group of order \(2k\). We calculate \(\sigma_c(D_{2k})\) and \(\Sigma_c(D_{2k})\), and we show that these two numbers coincide for any given \(c\) and \(k\).

Keywords: dihedral group; nilpotent cover; non-nilpotent subset

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Citation
Kimeu Arphaxad Ngwava, Nick Gill. Nilpotent Covers of Dihedral Groups[J], Ars Combinatoria, Volume 160. 7-12. DOI: https://www.doi.org/10.61091/ars-160-02.