The Groups of Isometries of Metric Spaces over Vector Groups

Sheng Bau and Yiming Lei
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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

Mathematics, 2022, vol. 10, issue 23, 1-9

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 , ? } where ? 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.

Keywords: abelian group; automorphism; isometry; vector group (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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