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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.

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Well...what have you done so far? From a first glance, I beieve you want to prove that you have a copy of $D_{\infty}$ acting on the $x$-axis and a second copy acting on the $y$-axis, and that these two subgroups generate your groups. Clearly they would intersect trivially and commute, so this would complete your proof...(to prove that these two subgroups exists, take the square that $W$ is acting on in an extended way and see what $W$ does to this square...basically, prove that the reflections of $D_4$ generate the entire groups!) –  user1729 Nov 14 '11 at 17:08

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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.

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