Exercise · ZW-DIHEDRAL-GROUP-008 · ★★☆ Intermediate
How many subgroups of order \(4\) does \(D_4\) have?
Choose one option.
Show answer
- A. 3
Explanation
There are three: the cyclic \(\langle r\rangle\) and the two Klein groups \(\{e,r^2,s,sr^2\}\) and \(\{e,r^2,sr,sr^3\}\).
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-009 Which group is D_3 isomorphic to?