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The Groups of Isometries of Metric Spaces over Vector Groups

Author

Listed:
  • Sheng Bau

    (School of Mathematics, Statistics and Computer Science, University of KwaZulu Natal, Pietermaritzburg 3209, South Africa)

  • Yiming Lei

    (School of Mathematics, Statistics and Computer Science, University of KwaZulu Natal, Pietermaritzburg 3209, South Africa
    College of Mathematical Sciences, Bohai University, Jinzhou 121013, China)

Abstract

In this paper, we consider the groups of isometries of metric spaces arising from finitely generated additive abelian groups. Let A be a finitely generated additive abelian group. Let R = { 1 , ϱ } where ϱ is a reflection at the origin and T = { t a : A → A , t a ( x ) = x + a , a ∈ A } . We show that (1) for any finitely generated additive abelian group A and finite generating set S with 0 ∉ S and − S = S , the maximum subgroup of Isom X ( A , S ) is R T ; (2) D ⊴ R T if and only if D ≤ T or D = R T ′ where T ′ = { h 2 : h ∈ T } ; (3) for the vector groups over integers with finite generating set S = { u ∈ Z n : | u | = 1 } , Isom X ( Z n , S ) = O n ( Z ) Z n . The paper also includes a few intermediate technical results.

Suggested Citation

  • Sheng Bau & Yiming Lei, 2022. "The Groups of Isometries of Metric Spaces over Vector Groups," Mathematics, MDPI, vol. 10(23), pages 1-9, November.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:23:p:4453-:d:984251
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    References listed on IDEAS

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    1. Dorin Andrica & Vasile Bulgarean, 2012. "Some Remarks on the Group of Isometries of a Metric Space," Springer Optimization and Its Applications, in: Panos M. Pardalos & Pando G. Georgiev & Hari M. Srivastava (ed.), Nonlinear Analysis, edition 127, chapter 0, pages 57-64, Springer.
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