Dihedral Groups as Epimorphic Images of Some Fibonacci Groups

The Fibonacci groups are defined by the presentation F (r,n)= (a1, a2, . . . , an : a1a2 · · · ar = ar+1, a2a3 . . . ar+1= ar+2, . . ., ana1 . . .ar-1=ar) where r 0 , n 0 and all subscripts are assumed to be reduced modulo n . In this paper we give an alternative proof that for r ≥ 0 , F(2r, 4r + 2) , F(4r + 3, 8r + 8) and F(4r + 5, 8r + 12) are all infinite by establishing a morphism (or group homomorphism) onto the dihedral group n D for all n 2 .