On the generalized order 2-Pell sequence of some classes of groups

ABSTRACT

In this paper, we study the 2-Pell number sequences modulo m. Also, by considering the finitely presented groups

${\displaystyle \begin{array}{ccc}\multicolumn{1}{c}{G_m}& =\hfill & ⟨a,b|a^m=b^m=1,{[a,b]}^a=[a,b],{[a,b]}^b=[a,b]⟩,m\ge 2,\hfill \\ \multicolumn{1}{c}{H_m}& =\hfill & ⟨a,b|a^{m^2}=b^m=1,b^{-1}ab=a^{1+m}⟩,m\ge 2,\hfill \end{array}}$

and

${\displaystyle \begin{array}{c}\multicolumn{1}{c}{K(n,l)=⟨a,b|a{b}^n=b^{l}a,b{a}^n=a^{l}b⟩,l>n>1,(n,l)=1,}\end{array}}$

we examine the lengths of the period of the generalized order 2-Pell sequences of these groups.

KEYWORDS:

• Period
• the generalized order 2-Pell sequence

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 20F05
• Secondary:11B39
• 20D60

Acknowledgment

The authors would like to thank the referee for helpful comments and suggestions.