Exercise · ZW-DIHEDRAL-GROUP-009 · ★★★ Advanced
Which group is \(D_3\) isomorphic to?
Choose one option.
Show answer
- C. \(S_3\)
Explanation
A triangle has \(6\) symmetries, which permute the three vertices and give every permutation: \(D_3\cong S_3\). The two other groups of order \(6\) are abelian and \(D_3\) is not.
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?