Dihedral group

The dihe­dral group \(D_{2n}\) is the group of sym­me­tries of the \(n\)-ver­tex reg­u­lar poly­gon.

Presentation

The dihe­dral groups have very sim­ple pre­sen­ta­tions:

$$D_{2n} \cong \langle a, b \mid a^n, b^2, b a b^{-1} = a^{-1} \rangle$$
The el­e­ment \(a\) rep­re­sents a ro­ta­tion, and the el­e­ment \(b\) rep­re­sents a re­flec­tion in any fixed axis. picture

Properties

  • The dihe­dral groups \(D_{2n}\) are all non-abelian for \(n > 2\). (Proof.)

  • The dihe­dral group \(D_{2n}\) is a sub­group of the sym­met­ric group \(S_n\), gen­er­ated by the el­e­ments \(a = (123 \dots n)\) and \(b = (2, n)(3, n-1) \dots (\frac{n}{2}+1, \frac{n}{2}+3)\) if \(n\) is even, \(b = (2, n)(3, n-1)\dots(\frac{n-1}{2}, \frac{n+1}{2})\) if \(n\) is odd.

Examples

\(D_6\), the group of sym­me­tries of the triangle

di­a­gram list the el­e­ments and Cayley table

In­finite dihe­dral group

The in­finite dihe­dral group has pre­sen­ta­tion \(\langle a, b \mid b^2, b a b^{-1} = a^{-1} \rangle\). It is the “in­finite-sided” ver­sion of the finite \(D_{2n}\).

We may view the in­finite dihe­dral group as be­ing the sub­group of the group of home­o­mor­phisms of \(\mathbb{R}^2\) gen­er­ated by a re­flec­tion in the line \(x=0\) and a trans­la­tion to the right by one unit. The trans­la­tion is play­ing the role of a ro­ta­tion in the finite \(D_{2n}\).

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Parents:

  • Group

    The alge­braic struc­ture that cap­tures sym­me­try, re­la­tion­ships be­tween trans­for­ma­tions, and part of what mul­ti­pli­ca­tion and ad­di­tion have in com­mon.