Photovoltaic Cells and Systems: SQU Journal for Science, 18 (2013) 54-59 © 2013 Sultan Qaboos University 54 Dihedral Groups as Epimorphic Images of Some Fibonacci Groups Abdullahi Umar a* and Bashir Ali b a Department of Mathematics and Statistics, Sultan Qaboos University,Al-Khod, PC 123 – Oman, *a Email: aumarh@squ.edu.om. b Department of Mathematics and Computer Science, Nigerian Defence Academy, Kaduna – Nigeria. ABSTRACT: The Fibonacci groups are defined by the presentation ),( nrF ,,,:,,, 213212121    rrrrn aaaaaaaaaaa , 11 rrn aaaa    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 0r  , (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 2n  . 1 Keywords: Group; Fibonacci group; Dihedral group; (homo) Morphism. مجموعات دايهيدرل كصورة متماثلة لمجموعات فيبوناتشى يبشير علو عبدهللا عمر انتمثيم بىاسطت تعسف مجمىعاث فيبىناتشيتعّسف ملخص : 1 2 1 2 3 1 2 1 1 ( , ) , ,..., : , , ... ,..., ... n r r n r r F r n a a a a a a a a a a a a       ,0 عندما تكىن 0n r  ، r r n a a   1نكم, ...,r n. 2) بأن، في هرا انبحث بسهانا بديلا نعطي , 4 2)F r r  ،( 4 3, 8 8)F r r  4)و 5, 8 12)F r r  في جمعيها ال منتهيت 0r حانت بىاسطت إيجاد دانت شمسة متماثهت وفىقيه عهى انصمسة وذنك n D 2 نكمn . : مجمىعاث ، مجمىعاث فيبىناتشي ، مجمىعاث دايهيدزل ، تشابه شكهي. مفتاح الكلمات 1. Introduction For 1r  and 1n  the Fibonacci group ( , )F r n is defined by the presentation: 1 2 1 2 1 2 3 1 2 1 1 ( , ) , , , : , , , n r r r r n r r F r n a a a a a a a a a a a a a a a         , 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:, 12 xyyxyxyxD n n   (1) It is well known that x and y in Dn satisfy the relations summarized in the next lemma. Lemma 1.1 For all 0 1k n   we have 1 MSC2010 : 20F05 FIBONACCI GROUPS 55 (a) ; knk xx   (b) ; 1 yy   (c) ;yxyx knk   (d) ;1)( 2 yx k (e) ;yyxx kk  (f) . knk xyyx   Thus we may write the elements of n D uniquely as k x or yx k for 0, 1, 2, , 1.k n  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 ),( nrF is indispensable for our discourse. Lemma 1.2 For all 0r  and 2m  we have 1 2 1 1m r m m r a a a       in ( , )F r n . Proof. 1211 1 111 )(     rmrmmmmmrmmmrm aaaaaaaaaa  . 2 1 1 1     rmm aa 2. Morphic Images First we consider the Fibonacci groups (2 , 4 2).F r r  Theorem 2.1 Let 0.r  There exist morphisms from (2 , 4 2)F r r  onto n D for all 3n  . Hence (2 , 4 2)F r r  is infinite. We are going to prove this theorem via a sequence of lemmas. However, we first define a mapping from the first 2r generators of (2 , 4 2)F r r  onto the generators of n D by xa x  and ( 2, 3, ... , 2 ). i a y i r (2) Then the next lemma gives the images of the remaining generators: .,,, 142212  rrr aaa  Lemma 2.2 (a) );1( 12   rxya r  (b) );1( 1 22    rxa n r  (c) );1( 2 32   ryxa r  (d) 2 ( 2 r i ra y   and 4 2 1).i r   Proof. Using Lemma 1.2 we see that (a) );1( 12 22112    rxyxyaaaa r rr  (b) );1()( 1212 12 1 122      rxxyxaaa n rr  (c) );1()( 22112 22 1 232      ryxxyaaa n rr  (d) This proof is by induction. Basis step: By Lemma 1.2 and (c) above, we see that .)( 2212 32 1 342 yyxyaaa rr       Inductive step: Suppose that ya ir 2 (for some 4 2i r  ). Using Lemma 1.2 again we see that 1 2 1 2 2 1 2 , r i i r i a a a y y y        ABDULLAHI UMAR and BASHIR ALI 55 as required. Lemma 2.3 For 1r  we have (a) );1(24  rxya r  (b) ;341 xaa r  (c) 2 4 (4 2 2). i r i i ra a y      Proof. Using Lemmas 1.2 and 2.2 we see that (a) ;)( 212 14 1 1224 xyyxyaaa rrr       (b) ;)( 22 24 1 2234 xxyxaaa rrr      (c) This proof is by induction. Basis step: By Lemma 1.2 and (b) above, we see that .)( 2122 34 1 3244 yxyxaaa rrr       Inductive step: Suppose that ya ir  4 (for some 4 2 1i r   ). Using Lemma 1.2 again we see that , 212 4 1 214 yyyaaa iririr       as required. 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).F r r  Theorem 2.4 Let .0r There exist morphisms from (4 3, 8 8)F r r  onto n D for all 3n  . Hence (4 3, 8 8)F r r  is infinite. As in the previous case, we are going to prove this theorem via a sequence of lemmas. First, we define a mapping from the first 34 r generators of (4 3, 8 8)F r r  onto the generators of n D by xaa ri  32 ,  and ,ya i  (3) where 32,342  riri and 0r . Then the next two lemmas give the images of the remaining generators: .,,, 885444  rrr aaa  Lemma 2.5 For 0r we have (a) ; 44 ya r   (b) ; 1 54   n r xa  (c) ; 2 64 yxa r   (d) 4 (7 2 6). r i i ra y     Proof . Using Lemma 1.2 we see that (a) 2 1 2 4 4 1 2 4 3 ; r r r r a a a a xy xy y      (b) 1 2 1 2 1 4 5 1 4 4 ; n r r a a a x y x        (c) ;)( 22112 54 1 264 yxxyaaa n rr       (d) This proof is by induction. Basis step: By Lemma 1.2 and (c) above, we see that 1 2 1 2 2 4 7 3 4 6 ( ) . r r a a a y x y y       Inductive step: Suppose that ya ir 4 (for some 7 2 5i r   ). Using Lemma 1.2 and the induction FIBONACCI GROUPS 55 hypothesis we see that , 212 4 1 314 yyyaaa iriir       as required. Lemma 2.6 For 0r we have (a) ; 1 76   n r xa  (b) ; 2 86 yxa r   (c) 6 (9 2 8). r i a y i r     Proof. Using Lemmas 1.2 and 2.5 we see that (a) ; 1212 66 1 3276      n rrr xyxaaa  (b) ;)( 22112 76 1 4286 yxxyaaa n rrr       (c) This proof is by induction. Basis step: For 9i  , we see tha .)( 2212 86 1 5296 yyxyaaa rrr       Inductive step: Suppose that ya ir  6 (for some 9 2 7i r   ). Then using Lemma 1.2, the fact that 9i  and induction hypothesis we see that , 212 6 1 )32(16 yyyaaa iririr       as required. Lemma 2.7 for 0r  we have (a) ; 981 xaa r    (b) 8 8 (10 2 10). i r i a a y i r       Proof. Using Lemmas 1.2, 2.5 and 2.6 we see that (a) ;)( 2112 88 1 5498 xyxaaa n rrr       (b) for 10 2 10,i r   we use induction. Basis step: For 10i  , we see that .)( 2122 1)88( 1 642)88( yxyxaaa rrr       Inductive step: Suppose that 8r i a y  (for some 10 2 9i r   ). Then using Lemma 1.2, (a) above and the induction hypothesis we see that 1 2 1 2 8 1 4 3 8 , r i r i r i a a a y y y          as required. Lemma 2.8 For 0r we have (a) ; )112(832 xaa rrr    (b) 8 8 (2 12 4 11). i r i r i ra a y       Proof. Using Lemmas 1.2, 2.6 and 2.7 we see that (a) ;)( 2112 )102(8 1 )72(4)112(8 xyxaaa n rrrrrr       (b) for 2 12 4 11r i r    we use induction. Basis step: For 2 12,i r  we see that .)( 2122 )112(8 1 )82(4)122(8 yxyxaaa rrrrrr       ABDULLAHI UMAR and BASHIR ALI 55 Inductive step: Suppose that ya ir  8 (for some 2 12 4 10r i r    ). Then using Lemma 1.2 (a) above and the induction hypothesis we see that , 212 8 1 3418 yyyaaa iririr       as required. 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 12)F r r  . Theorem 2.9 Let 0r . There exist morphisms from (4 5, 8 12)F r r  onto n D for all 3.n  Hence (4 5, 8 12)F r r  is infinite. 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 4 5r  generators of  4 5, 8 12F r r  onto the generators of nD by 1 2 3 , r a a x  and , i a y (4) where 32,542  riri and 0r . Analogously to Lemma 2.5 we have Lemma 2.10 For 0r  (a) ; 64 ya r   (b) ; 1 74   n r xa  (c) ; 2 84 yxa r   (d) ).829( 4   riya ir  Analogously to Lemma 2.6 we have Lemma 2.11 For 0r  we have. (a) ; 1 96   n r xa  (b) ; 2 106 yxa r   (c) 6 (11 2 12). r i i ra y     Analogously to Lemma 2.7 we have Lemma 2.12 For 0r  we have (a) ; 1381 xaa r    (b) 12 8 (14 2 14). i r i i ra a y      Analogously to Lemma 2.8 we have Lemma 2.13 For 0r  we have (a) ; )152(832 xaa rrr    (b) 12 8 (2 16 4 17). i r i r i ra a y       It is now clear from Lemmas 2.10, 2.11, 2.12 and 2.13 that the 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. FIBONACCI GROUPS 55 3. Acknowledgements Bashir Ali acknowledges the support and hospitality of Sultan Qaboos University in Spring 2011 during which period this work was completed. Bashir Ali also thanks the Nigerian Defence Academy for research leave and ETF Nigeria for financial support. We also thank the referees for helpful comments and suggestions, which greatly improved the exposition of this paper. 4. References CAMPBELL, C.M., DOOSTIE, H. and ROBERTSON, E.F. 2004. On the Fibonacci Length of Powers of Dihedral Groups. Applications of Fibonacci Numbers 9, Ed. F.T. Howard, Kluwer, Dordrecht, 69-85. CAMPBELL, C.M., ROBERTSON, E.F. and THOMAS, R.M. 1992. Fibonacci Groups and Semigroups. Technical Report CSD-50, Department of Computing Studies, University of Leicester. CONWAY, J.H. 1965. Solution of Advanced Problem 5327. American Mathematical Monthly 72(8): 915. GALLIAN, J.A. 1998. Contemporary Abstract Algebra, Houghton Mifflin, Boston/New York. THOMAS, R.M. 1991. The Fibonacci groups revisited. In Proceedings of Groups - St Andrews 1989, Volume 2 (London Math. Soc. Lecture Note Series 160, Cambridge University Press, 1991). (Eds.) CAMPBELL, C.M. and ROBERTSON, E.F. 445-454. Received 12 April 2011 Accepted 16 June 2012