Show Summary Details
More options …

# Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board: Ciobanu, Laura / Conder, Marston / Eick, Bettina / Elder, Murray / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Perret, Ludovic / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura / Lohrey, Markus

CiteScore 2018: 0.80

SCImago Journal Rank (SJR) 2018: 0.368
Source Normalized Impact per Paper (SNIP) 2018: 1.061

Mathematical Citation Quotient (MCQ) 2018: 0.38

Online
ISSN
1869-6104
See all formats and pricing
More options …
Volume 7, Issue 1

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

Anthony M. Gaglione
/ Seymour Lipschutz
/ Dennis Spellman
Published Online: 2015-03-17 | DOI: https://doi.org/10.1515/gcc-2015-0004

## 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 ${G}^{ℤ\left[\theta \right]}$ of 2×2 matrices admitting exponents from the integral polynomial ring $ℤ\left[\theta \right]$. Identifying G with its matrix representation, we show that given $\gamma \left(\theta \right)\in {G}^{ℤ\left[\theta \right]}$ and $n\in ℤ$, one has that ${lim}_{\theta \to n}\gamma \left(\theta \right)$ exists and lies in G. Furthermore, the maps $\gamma \left(\theta \right)↦{lim}_{\theta \to n}\gamma \left(\theta \right)$ form a discriminating family of group retractions ${G}^{ℤ\left[\theta \right]}\to G$ 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

Revised: 2014-05-31

Published Online: 2015-03-17

Published in Print: 2015-05-01

Citation Information: Groups Complexity Cryptology, Volume 7, Issue 1, Pages 59–68, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144,

Export Citation