A class of Z-metacyclic groups involving the Lucas numbers

H. Doostie, K. Ahmadidelir

The sequence $\{g_i\}^\infty_{i=1}$ is the sequence of Lucas numbers $g_1=2$, $g_2=1$, $g_{i+2}=g_{i+1}+gi$, $(i\geq1)$, and $\ell\geq2$ is an integer. In this paper we consider the group $G(\ell)$ with an efficient presentation $\langle x,y\mid x^\ell=y^\ell=xyx^{[\frac\ell2]}y^{[\frac{3\ell}2]}\rangle$ where, $[x]$ is used for the integer part of a real $x$, and prove that $G(\ell)$ is finite of order \[ |G(\ell)|=\begin{cases} \frac{\ell(\ell+2)}2(1+3^{\frac\ell2}), & \ell\equiv0ext{ or }m2(\mod6) 2\ell(\ell+1)g_{\ell+1}, &\ell\equiv3(\mod6), \ell(\ell+1)g_{\ell+1}, & \ell\equivm1(\mod6). \end{cases} \] Moreover, if $\ell=\pm4$ or 8 or $\pm12$ or $20(\mod40)$, or $\ell\equiv\pm1(\mod 6)$ then, $G(\ell)$ is Z-metacyclic ($G'(\ell)$ and $\frac{G(\ell)}{G'(\ell)}$ are cyclic).