Tile the Euclidean plane by squares of side length 1. Let W be the group generated by the four reflections in the (extended) sides of any one square. Draw the Cayley graph of W and prove that $W = D_\infty \bigoplus D_\infty$. I am so confused with this question. Any help would be great to get started.
2026-03-28 19:37:48.1774726668
Tiling the Euclidean plane, and Cayley graph
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What may be confusing about this is that the role that's usually played by rotations in a dihedral group is played by translations here. However, abstractly, this is the same thing: The translations along an axis form an infinite cyclic group, just like the one generated by a rotation through an irrational multiple of $\pi$ in the "normal" infinite dihedral group; and conjugation by any reflection inverts a translation, which is again the same as for the generators of dihedral groups. You can take one reflection and one translation as generators for each axis.