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Let $G$ be the set of bijections $\mathbb{R} \to \mathbb{R}$ which preserve the distance between pairs of points, and send integers to integers. Then $G$ is a group under composition of functions. The following two elements are obviously in $G$: the function $t$ (translation) where $t(x)=x+1$ for each $x \in \mathbb{R}$ and the function $r$ (reflection) where $r(x)=-x$ for each $x \in \mathbb{R}$. The subgroup of $G$ generated by $r$ and $t$ is called the infinite dihedral group and denoted by $D_{\infty}$. Note that this information describes an action of $D_{\infty}$ on $\Re$.

  1. Show that every element of $D_{\infty}$ can be written uniquely in the form $r^it^j$ for suitably restricted values of $i$ and $j$. Explain how to multiply two such elements.

  2. Describe geometrically the possible sorts of actions of elements of $D_{\infty}$ on the real line $\mathbb{R}$.

I have shown that every element is of the form $r^it^j$ but I am unsure how two elements would be multiplied together and how the actions should be described geometrically.

Can anybody help?

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