Non-Abelian Sequenceable Groups Involving A-Covers

A non-abelian finite group g is called sequenceable if for some positiveinteger k , g is k -generated (g =< α1α 2 , ... ,α k » and there exist integersa α 1 , ... , αk such that every element of g is a term of the k -step generalized fibonacci sequence xi =αi ,i=1,2, ... .k , xi =(xi-k)^α1 (xi-k+l)^α2 ... (xi_1)^αk,i >k +1. a remarkable application of this definition may be find on the study ofrandom covers in the cryptography. the 2-step generalized sequences for thedihedral groups studied for their periodicity in 2006 by h. aydin and it is provedthat in many cases for a j and a 2 , they are not periodic. aydin's work was incontinuation of the research works of r. dikici (1997) and e. ozkan (2003) wherethey studied the ordinary fibonacci sequences (sequences without the powers) ofelements of groups. in this paper we consider 3-step generalized fibonaccisequences and prove that the quatemion group q2 (for every integer n ≥3) andthe dihedral group d 2n (for every integer n≥ 3 ) are sequenceable. the α -covers together with the fibonacci lengths of the corresponding 3-step sequences havebeen calculated as well.