An application of elementary real analysis to a metabelian group admitting integral polynomial exponents

Anthony M. Gaglione; Seymour Lipschutz; Dennis Spellman

doi:10.1515/gcc-2015-0004

Article

An application of elementary real analysis to a metabelian group admitting integral polynomial exponents

Anthony M. Gaglione , Seymour Lipschutz and Dennis Spellman

Published/Copyright: March 17, 2015

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From the journal Groups Complexity Cryptology Volume 7 Issue 1

Abstract

Let G be a free metabelian group of rank r = 2. We introduce a faithful 2×2 real matrix representation of G and extend this to a group Gℤ[θ] of 2×2 matrices admitting exponents from the integral polynomial ring ℤ[θ]. Identifying G with its matrix representation, we show that given γ(θ)∈Gℤ[θ] and n∈ℤ, one has that limθ→nγ(θ) exists and lies in G. Furthermore, the maps γ(θ)↦limθ→nγ(θ) form a discriminating family of group retractions Gℤ[θ]→G as n varies over ℤ. Although not explicitly carried out in this manuscript, it is clear that similar results hold for any countable rank r.

Keywords: Metabelian group; exponential group

MSC: 20E26; 03C07; 20F19; 20F05

Received: 2014-2-8

Revised: 2014-5-31

Published Online: 2015-3-17

Published in Print: 2015-5-1

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https://doi.org/10.1515/gcc-2015-0004

Keywords for this article

Metabelian group; exponential group