\(k\)-nacci sequences in finite triangle groups.
A \(k\)-sequence in a finite group is a sequence of group elements \(x_0, x_1, \dots\) for which the \(j\)-element set \(\{x_0, x_1, \dots, x_{j-1} \}\) is fixed and each element \(x_n\) is defined by \[ x_n = \begin{cases} x_0x_1\dots x_{n-1}, & \text{for } j \leq n < k \\ x_{n-k}x_{n-k+1}\dots x_{n-1}, & \text{for } n \geq k \end{cases} \] The polyhedral group \((l,m,n)\) for \(l, m, n > 1\) is defined by the expression \[ \langle x,y,z: x^l = y^m = z^n = xyz = e \rangle. \] Some properties of these groups are discussed. For example, Theorem 2.2: Let \(G_2\) be the group defined by the expression \[ G_2 = \langle x,y,z: x^2 = y^2 = z^2 = xyz = e \rangle. \] Then its period is \(k+1\). Theorem 2.3: Let \(G_n\), \(n > 2\) be the group defined by the expression \[ G_n = \langle x,y,z: x^n = y^2 = z^2 = xyz = e \rangle. \] Then its period is \(2k+2\). The extended triangle group \(E(p,q,r)\) for \(p,q,r > 1\) is defined by the presentation \[ \langle x,y,z: x^2 = y^2 = z^2 = (xy)^p = (yz)^q = (zx)^r = e \rangle. \] Theorem 3.2: Let \(E_2\) be the group defined by the expression \[ E_2 = \langle x,y,z: x^2 = y^2 = z^2 = (xy)^2 = (yz)^2 = (zx)^2 = e \rangle. \] Then its period is \(k+1\). An extension of this theorem is formulated and proved.
scientific article; zbMATH DE number 5637362
scientific article