Combinatorics Through Guided Discovery, 2004a
Combinatorics Through Guided Discovery, 2004a
Combinatorics Through Guided Discovery, 2004a
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6.1. Permutation Groups 109<br />
Problem 262. How many elements does the group D n have? Prove that you<br />
are correct.<br />
Problem 263. In Figure 6.1.3 we show a cube with the positions of its vertices<br />
and faces labeled. As with motions of the square, we let ϕ(x) be the label<br />
of the place where vertex previously in position x is now.<br />
Figure 6.1.3: A cube with the positions of its vertices and faces labelled. The<br />
curved arrows point to the positions that are blocked by the cube.<br />
(a) Write in two row notation the permutation ρ of the vertices that corresponds<br />
to rotating the cube 90 degrees around a vertical axis through<br />
the faces t (for top) and u (for underneath). (Rotate in a right-handed<br />
fashion around this axis, meaning that vertex 6 goes to the back and<br />
vertex 8 comes to the front.)<br />
(b) Write in two row notation the permutation ϕ that rotates the cube<br />
120 degrees around the diagonal from vertex 1 to vertex 7 and carries<br />
vertex 8 to vertex 6.<br />
(c) Compute the two row notation for ρ ◦ ϕ<br />
(d) Is the permutation ρ ◦ ϕ a rotation of the cube around some axis? If<br />
so, say what the axis is and how many degrees we rotate around the<br />
axis. If ρ ◦ ϕ is not a rotation, give a geometic description of it.<br />
⇒ ·<br />
Problem 264. How many permutations are in the group R? R is sometimes<br />
called the “rotation group” of the cube. Can you justify this? (h)
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