Exercise · ZW-DIHEDRAL-GROUP-010 · ★★★ Advanced
In \(D_n\), what does \(r^{k}s\) equal?
Choose one option.
Show answer
- B. \(sr^{-k}\)
Explanation
From \(srs=r^{-1}\) we get \(sr=r^{-1}s\), and repeating, \(sr^k=r^{-k}s\). Replacing \(k\) by \(-k\) gives \(r^{k}s=sr^{-k}\).
Review the concept: Dihedral group
The dihedral group \(D_n\), with \(n\ge3\), is the group of symmetries of a regular \(n\)-gon, generated by the rotation \(r\) by \(2\pi/n\) and a reflection \(s\).
\[D_n=\langle r,s\mid r^n=s^2=e,\ srs=r^{-1}\rangle\]
More practice problems: dihedral group
- ZW-DIHEDRAL-GROUP-001 How many elements does D_6 have?
- ZW-DIHEDRAL-GROUP-002 In D_7, what is the order of the rotation r by 2π/7?
- ZW-DIHEDRAL-GROUP-003 How many reflections does D_5 have?
- ZW-DIHEDRAL-GROUP-004 In D_n with the usual presentation, what does srs equal?
- ZW-DIHEDRAL-GROUP-005 How many elements of order 2 does D_4 have?
- ZW-DIHEDRAL-GROUP-006 What is the order of the center of D_6?
- ZW-DIHEDRAL-GROUP-007 In D_8, what is the order of r^6?
- ZW-DIHEDRAL-GROUP-008 How many subgroups of order 4 does D_4 have?