Problem

\(D_{2n}=\langle r,s | r^n = s^2 = 1, rs = sr^{-1}\rangle\).

Compute the order of each of the elements in the following groups:

• \(D_6\)
• \(D_8\)
• \(D_{10}\)

Solution

\(D_6\)

• \(|1|=1\)
• \(|r| = 3\)
• \(|r^2| = 3\)
• \(|s| = 2\)
• \(|sr| = 2\)
• \(|sr^2| = 2\)

\(D_8\)

• \(|1|=1\)
• \(|r| = 4\)
• \(|r^2| = 2\)
• \(|r^3| = 4\)
• \(|s| = 2\)
• \(|sr| = 2\)
• \(|sr^2| = 2\)
• \(|sr^3| = 2\)

\(D_{10}\)

• \(|1|=1\)
• \(|r| = 5\)
• \(|r^2| = 5\)
• \(|r^3| = 5\)
• \(|r^4| = 5\)
• \(|s| = 2\)
• \(|sr| = 2\)
• \(|sr^2| = 2\)
• \(|sr^3| = 2\)
• \(|sr^4| = 2\)

$$\tag*{$\blacksquare$}$$