If $E=\mathbb{R}\times (-1,1)$ and $\forall (x,t), (y,t^{'}) \in \mathbb{R}\times (-1,1)$, $$(x,t)\backsimeq(y,t^{'})\Leftrightarrow \exists n\in \mathbb{Z} ; x=y+2n\pi , t=(-1)^{n} t^{'} $$ $$x\backsim y \Leftrightarrow \exists n\in \mathbb{Z} ; x=y+2n\pi$$ $S^{1}= \mathbb{R}/ \backsim$ and $E= \mathbb{R}\times (-1,1)/\backsimeq$
$E$ is called Möbius band.
Now, I would like to investigate its trivializations and structure group.
Suppose $\pi: E\to S^{1}$ that $[x,t]\mapsto [x]$ and $U_{1}=(0,2\pi)/\backsim, U_{2}=(-\pi,\pi)/\backsim$ and $(\pi, \phi_{i}):\pi^{-1}(U_{i})\to U_{i}\times \mathbb{R}$ that for all $i\in\{1,2\}$, $[x,t]\mapsto([x],t)$.
Translation map is in the following form $$(\pi, \phi_{2})\circ (\pi, \phi_{1})^{-1}:(U_{1}\cap U_{2})\times \mathbb{R}\to (U_{1}\cap U_{2})\times \mathbb{R}$$,
$$([x],t)\mapsto [x,t]=[x,t]\quad if\quad x\in (0,\pi)\quad and \quad [x,t]=[x-2\pi,-t]\quad if \quad x\in (\pi, 2\pi)$$ Now, using $(\pi, \phi_{2})$, we have: $$([x],t)\mapsto t \quad if\quad x\in (0,\pi)\quad and \quad ([x],t)\mapsto -t\quad if \quad x\in (\pi, 2\pi)$$ Is it correct?
Why does it have structure group $G=\mathbb{Z}_{2 }$?
Help me , please.