\(k\)-nacci sequences in finite triangle groups.
type: Item
instance of: scholarly article
title: \(k\)-nacci sequences in finite triangle groups.
zbMATH Open document ID: 1197.11025
DOI: 10.1155/2009/453750
author: Erdal Karaduman
author: Ömür Deveci
published in: Discrete Dynamics in Nature and Society
publication date: 2009-11-23T00:00:00Z
full work available at URL: https://eudml.org/doc/229786
review text: A \(k\)-sequence in a finite group is a sequence of group elements \(x_0, x_1, \dots\) for which the \(j\)-element set \(\{x_0, x_1, \dots, x_{j-1} \}\) is fixed and each element \(x_n\) is defined by \[ x_n = \begin{cases} x_0x_1\dots x_{n-1}, & \text{for } j \leq n < k \\ x_{n-k}x_{n-k+1}\dots x_{n-1}, & \text{for } n \geq k \end{cases} \] The polyhedral group \((l,m,n)\) for \(l, m, n > 1\) is defined by the expression \[ \langle x,y,z: x^l = y^m = z^n = xyz = e \rangle. \] Some properties of these groups are discussed. For example, Theorem 2.2: Let \(G_2\) be the group defined by the expression \[ G_2 = \langle x,y,z: x^2 = y^2 = z^2 = xyz = e \rangle. \] Then its period is \(k+1\). Theorem 2.3: Let \(G_n\), \(n > 2\) be the group defined by the expression \[ G_n = \langle x,y,z: x^n = y^2 = z^2 = xyz = e \rangle. \] Then its period is \(2k+2\). The extended triangle group \(E(p,q,r)\) for \(p,q,r > 1\) is defined by the presentation \[ \langle x,y,z: x^2 = y^2 = z^2 = (xy)^p = (yz)^q = (zx)^r = e \rangle. \] Theorem 3.2: Let \(E_2\) be the group defined by the expression \[ E_2 = \langle x,y,z: x^2 = y^2 = z^2 = (xy)^2 = (yz)^2 = (zx)^2 = e \rangle. \] Then its period is \(k+1\). An extension of this theorem is formulated and proved.
Mathematics Subject Classification ID: 11B39
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 20F12
zbMATH DE Number: 5637362
Wikidata QID: Q58647481
reviewed by: Krassimir Atanassov
MaRDI profile type: Publication
OpenAlex ID: W2144749783
cites work: Q3972195
cites work: Fibonacci Series Modulo m
cites work: Q4657461
cites work: Fibonacci length of automorphism groups involving Tribonacci numbers
cites work: Q4936011
cites work: Finite p -Quotients of Some Cyclically Presented Groups
cites work: On the relationship between the recurrences in nilpotent groups and the binomial formula
cites work: The Fibonacci length of certain centro-polyhedral groups.
cites work: Fibonacci length of direct products of groups
cites work: Fibonacci lengths involving the Wall number \(k(n)\).
cites work: On truncated Fibonacci sequences
cites work: Q3739201
cites work: Q4007084
cites work: The Fibonacci lengths of binary polyhedral groups and related groups.
cites work: Q5673083
cites work: Q4463885
cites work: The triangle groups
description: scientific article; zbMATH DE number 5637362