Furkan BİROL, Özden KORUOĞLU, Bilal DEMİR

Genişletilmiş modüler grubun H ̅_3,3 alt grubu ve Fibonacci sayıları

Bu çalışmada tam katsayılı otomorfizma ve anti-otomorfizmaların grubu olan genişletilmiş modüler grubun H ̅_3,3 alt grubunun elemanları ve Fibonacci sayıları arasındaki ilişki incelenecektir. H ̅_3,3 grubu, altı mertebeli iki dihedral grubun iki mertebeli devirli grup altında, birleştirilmiş serbest çarpımıdır. Başka bir ifadeyle bu grup üç mertebeli X(z)=(-1)⁄((z-1) ), Y(z)=(-1)⁄((z+1) ) şeklinde iki eleman ve iki mertebeli R(z)=1⁄z ̅ bir yansıma dönüşümü tarafından üretilir. Böylece grubun herbir elemanı X, Y ve R nin kuvvetlerinin bir kombinasyonu olarak yazılabilir. Burada bu kombinasyonun içinde katsayıları Fibonacci sayıları olan bloklar elde edilecektir.

The subgroup H ̅_3,3 of extended modular group and Fibonacci numbers

In this study, relationships between the elements of the group H ̅_3,3, subgroup of extended modular group that consists of automorphisms and anti-automorphisms with integer coefficients, and Fibonacci numbers are studied. The group H ̅_3,3 is isomorphic to amalgamated free product of two dihedral groups of order six with cyclic group of order two. In other words this group is generated by two elements X(z)=(-1)⁄((z-1) ), Y(z)=(-1)⁄((z+1) ) of order three and a two ordered reflection R=1⁄z ̅ . Hence every element of H ̅_3,3 can be written as a combination of powers of X,Y and R. Here, it is investigated the blocks in this combination whose coefficients are Fibonacci numbers.

Keywords:

Fibonacci numbers, modular group, extended generalized Hecke groups,

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