Exercise · ZW-DIHEDRAL-GROUP-004 · ★☆☆ Basic
In \(D_n\) with the usual presentation, what does \(srs\) equal?
Choose one option.
Show answer
- C. \(r^{-1}\)
Explanation
The defining relation is \(srs=r^{-1}\): conjugating a rotation by a reflection inverts it.
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-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?
- ZW-DIHEDRAL-GROUP-009 Which group is D_3 isomorphic to?