Dihedral Groups as Epimorphic Images of Some Fibonacci Groups

The Fibonacci groups are defined by the presentation  ) , ( n r F , , , : , , , 2 1 3 2 1 2 1 2 1          r r r r n a a a a a a a a a a a , 1 1 r r n a a a a    where 0  r , 0  n and all subscripts are assumed to be reduced modulo n . In this paper we give an alternative proof that for 0 r  , (2 , 4 2) F r r  , (4 3, 8 8) F r r   and (4 5, 8 12) F r r   are all infinite by establishing a morphism (or group homomorphism) onto the dihedral group n D for all 2 n  . 1


ABSTRACT:
The Fibonacci groups are defined by the presentation

Introduction
For 1 r  and 1 n  the Fibonacci group ( , )  F r n is defined by the presentation: , where all subscripts are assumed to be reduced modulo n , if necessary.These groups were first introduced by Conway (1965) and have been studied over the last few decades.For a nice survey article see (Thomas, 1991) or (Campbell et al., 1992).
The dihedral group of order 2n denoted by n D is usually defined by ., 1 : , It is well known that x and y in D n satisfy the relations summarized in the next lemma.
Lemma 1.1 For all 01 kn    we have (a) ; Thus we may write the elements of n D uniquely as k x or y x k for 0, 1, 2, , 1. kn  Campbell et al. (2004) explored the connection between the Fibonacci groups and finite groups via the concept of Fibonacci length.In the case where the finite groups were dihedral they obtained satisfactory results.In this note we further explore the connection between the Fibonacci groups and dihedral groups in a different manner.In particular, we establish epimorphisms between Fibonacci groups in certain classes and all finite dihedral groups of order greater than 4, thus giving alternative proofs regarding the infiniteness of the groups in these classes of Fibonacci groups.For basic concepts in group theory we refer the reader to (Gallian, 1998).The following lemma for
Then the next lemma gives the images of the remaining generators: ., , , (for some 42 ir  ).Using Lemma 1.2 again we see that (4 2 2).(for some 4 2 1 ir    ).Using Lemma 1.2 again we see that , It is now clear from Lemmas 2.2 and 2.3 that the mapping defined in ( 2) is indeed a morphism onto n D , which preserves all the relations of (2 , 4 2) F r r  and so Theorem 2.1 is proved.
Next we consider the Fibonacci groups (4 3, 8 8).; ; (for some 9 2 7 ir    ).Then using Lemma 1.2, the fact that 9 i  and induction hypothesis we see that ,  we use induction.

Basis step:
ay  (for some 10 2 9 ir    ).Then using Lemma 1.2, (a) above and the induction hypothesis we see that It is now clear from Lemmas 2.5, 2.6, 2.7 and 2.8 that the mapping defined in (3) is indeed a morphism onto , n D which preserves all the relations of (4 3, 8 8) F r r  and so Theorem 2.4 is proved.
Finally we consider the Fibonacci groups (4 5, 8  As in the previous cases, we are going to prove this theorem via a sequence of lemmas.However, since the proofs are similar to the previous case we are going to state the corresponding results without proofs.We first define a mapping from the first 45 r  generators of     (2 16 4 17).


It is now clear from Lemmas 2.10, 2.11, 2.12 and 2.13 that mapping defined in ( 4) is indeed a morphism onto n D , which preserves all the relations of (4 5, 8 12) F r r  and so Theorem 2.9 is proved.
previous case, we are going to prove this theorem via a sequence of lemmas.First, we define a This proof is by induction.
This proof is by induction.