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Section5.2Dihedral Groups

Another special type of permutation group is the dihedral group. Recall the symmetry group of an equilateral triangle in Chapter 3. Such groups consist of the rigid motions of a regular \(n\)-sided polygon or \(n\)-gon. For \(n = 3, 4, \ldots\), we define the nth dihedral group to be the group of rigid motions of a regular \(n\)-gon. We will denote this group by \(D_n\). We can number the vertices of a regular \(n\)-gon by \(1, 2, \ldots, n\) (Figure 5.19). Notice that there are exactly \(n\) choices to replace the first vertex. If we replace the first vertex by \(k\), then the second vertex must be replaced either by vertex \(k+1\) or by vertex \(k-1\); hence, there are \(2n\) possible rigid motions of the \(n\)-gon. We summarize these results in the following theorem.

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Figure5.19A regular \(n\)-gon

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Figure5.21Rotations and reflections of a regular \(n\)-gon

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Figure5.22Types of reflections of a regular \(n\)-gon
Proof
Example5.24

The group of rigid motions of a square, \(D_4\), consists of eight elements. With the vertices numbered 1, 2, 3, 4 (Figure 5.25), the rotations are \begin{align*} r & = (1234)\\ r^2 & = (13)(24)\\ r^3 & = (1432)\\ r^4 & = (1) \end{align*} and the reflections are \begin{align*} s_1 & = (24)\\ s_2 & = (13). \end{align*} The order of \(D_4\) is 8. The remaining two elements are \begin{align*} r s_1 & = (12)(34)\\ r^3 s_1 & = (14)(23). \end{align*}

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Figure5.25The group \(D_4\)

SubsectionThe Motion Group of a Cube

We can investigate the groups of rigid motions of geometric objects other than a regular \(n\)-sided polygon to obtain interesting examples of permutation groups. Let us consider the group of rigid motions of a cube. One of the first questions that we can ask about this group is “what is its order?” A cube has 6 sides. If a particular side is facing upward, then there are four possible rotations of the cube that will preserve the upward-facing side. Hence, the order of the group is \(6 \cdot 4 = 24\). We have just proved the following proposition.

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Figure5.26The motion group of a cube
Proof

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Figure5.29Transpositions in the motion group of a cube