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Paper Information

Journal:   JOURNAL OF SCIENCES ISLAMIC REPUBLIC OF IRAN   SUMMER 2009 , Volume 20 , Number 3; Page(s) 277 To 282.
 
Paper: 

NON-ABELIAN SEQUENCEABLE GROUPS INVOLVING α-COVERS

 
 
Author(s):  SADEGHIEH A., DOUSTI H.*
 
* MATHEMATICS DEPARTMENT, FACULTY OF MATHEMATICS AND COMPUTER SCIENCES, TARBIAT MOALLEM UNIVERSITY, 49 MOFATEH AVE., TEHRAN 15614, ISLAMIC REPUBLIC OF IRAN
 
Abstract: 

A non-abelian finite group G is called sequenceable if for some positive integer k, G is k -generated (G=<a1, a2, ..., ak >) and there exist integers a1, a2, ... , ak such that every element of G is a term of the k -step generalized Fibonacci sequence xi=ai, i=1, 2, ... , k, xi=(xi-k)a1 (xi-k+1)a2 (xi-1)ak, i³k+1. A remarkable application of this definition may be find on the study of random covers in the cryptography. The 2-step generalized sequences for the dihedral groups studied for their periodicity in 2006 by H. Aydin and it is proved that in many cases for a1 and a2, they are not periodic. Aydin’s work was in continuation of the research works of R. Dikici (1997) and E. Ozkan (2003) where they studied the ordinary Fibonacci sequences (sequences without the powers) of elements of groups. In this paper we consider 3-step generalized Fibonacci sequences and prove that the quaternion group Q2n (for every integer n³3) and the dihedral group D2n (for every integer n³3) are sequenceable. The a-covers together with the Fibonacci lengths of the corresponding 3-step sequences have been calculated as well.

 
Keyword(s): FIBONACCI LENGTH, FINITE GROUPS
 
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