Consider the group of isometries $N'$ of an infinite ribbon $$R=\left\{(x,y)| -1 \le y \le 1\right\}.$$ The following elements are in $N'$: $$t_a:(x,y)\to(x+a,y)$$ $$s:(x,y)\to(-x,y)$$ $$r:(x,y)\to(x,-y)$$ $$\rho:(x,y)\to(-x,-y)$$ I have shown that they satisfy the following relations: $$t_at_b=t_{a+b}$$ $$s^2=r^2=\rho^2=1$$ $$st_a=t_{-a}s$$ $$rt_a=t_ar$$ $$\rho t_a=t_{-a}\rho$$ $$rs=sr=\rho$$ $$s\rho=\rho s=r$$ $$r\rho=\rho r=s$$ Then I am asked to consider the frieze pattern, which is a pattern on the ribbon that is periodic and whose group of symmetries is discrete.
Does it mean I only need to restrict the $a$ to integer multiples of the period of the pattern, and all is done?