Groups Related to F a,b,c Involving Fibonacci Numbers

Abstract

The definition of the groups $$ {F^{a,b,c}}\;{\rm{ = }}\left\langle {\left. {x,y|{x^2}\;{\rm{ = }}1, x{y^a}x{y^b}x{y^c}\;{\rm{ = 1}}} \right\rangle } \right. $$ was suggested by H. S. M. Coxeter because of their relevance to the search for trivalent 0-symmetric Cayley diagrams, and these groups are studied in [1]. The structure of these groups depends heavily on that of the groups $$ {H^{a,b,c}}\;{\rm{ = }}\left\langle {\left. {x,y|\;{x^2} = 1, {y^{2n}}\;{\rm{ = 1, }}x{y^a}x{y^b}x{y^c}\;{\rm{ = 1}}} \right\rangle ,} \right. $$ where n = a + b + c. The groups H a,b,c and related groups are studied in [1] and [2]. Perhaps rather surprisingly, the orders of the groups F a,b,c and H a,b,c sometimes involve Fibonacci numbers; see for example Theorem 9.1 of [1]. Another connection with Fibonacci numbers emerges from a study of the apparently unrelated class of groups discussed in [3]. In the work of this last paper the groups T(2m) = 〈x,t|xt 2 xtx 2 t = 1, xt 2m + 1 = t 2 x 2)〉 are shown to have order \( \mathop {40mf}\nolimits_m^3 \;{\rm{if }}m is odd and \mathop {8mg}\nolimits_m^3 \) if m is even, where f m and g m are Fibonacci and Lucas numbers respectively. (Definitions and useful properties of Fibonacci and Lucas numbers may be found in [3], [5], [8].) To see that the groups T(2m) are in fact closely related to the groups F a,b,c , put a = xt, b = t to obtain $$ T\left( {2m} \right)\;{\rm{ = }}\left\langle {a,b|{b^{ - 1}}{a^2}b = {a^{ - 2}},aba{b^{ - 2}}a{b^{2m + 1}} = 1} \right\rangle . $$ Then 〈a 2〉 is normal in T(2m), and the factor T(2m)/〈a 2〉 is isomorphic to F 1,−2,2m+ 1.