Journal of Applied Mathematics
Volume 2012 (2012), Article ID 464580, 15 pages
http://dx.doi.org/10.1155/2012/464580
Research Article

The Generalized Order- π‘˜ Lucas Sequences in Finite Groups

Γ–mΓΌr Deveci1,2 and Erdal Karaduman1,2

1Department of Mathematics, Faculty of Arts and Science, Kafkas University, 36100 Kars, Turkey
2Department of Mathematics, Faculty of Science, AtatΓΌrk University, 25240 Erzurum, Turkey

Received 18 October 2011; Accepted 11 December 2011

Academic Editor: Reinaldo Martinez Palhares

Copyright © 2012 Ömür Deveci and Erdal Karaduman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the generalized order- π‘˜ Lucas sequences modulo π‘š . Also, we define the 𝑖 th generalized order- π‘˜ Lucas orbit 𝑙 𝑖 , { 𝛼 1 , 𝛼 2 , … , 𝛼 π‘˜ βˆ’ 1 } 𝐴 ( 𝐺 ) with respect to the generating set 𝐴 and the constants 𝛼 1 , 𝛼 2 , and 𝛼 π‘˜ βˆ’ 1 for a finite group 𝐺 = ⟨ 𝐴 ⟩ . Then, we obtain the lengths of the periods of the 𝑖 th generalized order- π‘˜ Lucas orbits of the binary polyhedral groups ⟨ 𝑛 , 2 , 2 ⟩ , ⟨ 2 , 𝑛 , 2 ⟩ , ⟨ 2 , 2 , 𝑛 ⟩ and the polyhedral groups ( 𝑛 , 2 , 2 ) , ( 2 , 𝑛 , 2 ) , ( 2 , 2 , 𝑛 ) for 1 ≀ 𝑖 ≀ π‘˜ .