The Adjacency-Jacobsthal Sequence in Finite Groups

Abstract

The adjacency-Jacobsthal sequence and the adjacency-Jacobsthal matrix were defined by Deveci and Artun (see [5]). In this work, we consider the cyclic groups which are generated by the multiplicative orders of the adjacency-Jacobsthal matrix when read modulo α (α > 1). Also, we study the adjacency-Jacobsthal sequence modulo α and then we obtain the relationship among the periods of the adjacency-Jacobsthal sequence modulo α and the orders of the cyclic groups obtained. Furthermore, we redefine the adjacency-Jacobsthal sequence by means of the elements of 2-generator groups which is called the adjacency-Jacobsthal orbit. Then we examine the adjacency-Jacobsthal orbit of the finite groups in detail. Finally, we obtain the periods of the adjacency-Jacobsthal orbit of the dihedral group D₁₀ as applications of the results obtained.

Key words and phrases. The Adjacency-Jacobsthal Sequence, Group, Matrix, Period.

1991 Mathematics Subject Classification. 11K31, 20F05, 11C20