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Section6.5Dihedral groups

Dihedral groups are groups of symmetries of regular \(n\)-gons. We start with an example.


Consider a regular triangle \(T\text{,}\) with vertices labeled \(1\text{,}\) \(2\text{,}\) and \(3\text{.}\) We show \(T\) below, also using dotted lines to indicate a vertical line of symmetry of \(T\) and a rotation of \(T\text{.}\)

<<SVG image is unavailable, or your browser cannot render it>>

Note that if we reflect \(T\) over the vertical dotted line (indicated in the picture by \(f\)), \(T\) maps onto itself, with \(1\) mapping to \(1\text{,}\) and \(2\) and \(3\) mapping to each other. Similarly, if we rotate \(T\) clockwise by \(120^{\circ}\) (indicated in the picture by \(r\)), \(T\) again maps onto itself, this time with \(1\) mapping to \(2\text{,}\) \(2\) mapping to \(3\text{,}\) and \(3\) mapping to \(1\text{.}\) Both of these maps are called symmetries of \(T\text{;}\) \(f\) is a reflection or flip and \(r\) is a rotation.

Of course, these are not the only symmetries of \(T\text{.}\) If we compose two symmetries of \(T\text{,}\) we obtain a symmetry of \(T\text{:}\) for instance, if we apply the map \(f\circ r\) to \(T\) (meaning first do \(r\text{,}\) then do \(f\)) we obtain reflection over the line connecting \(2\) to the midpoint of line segment \(\overline{13}\text{.}\) Similarly, if we apply the map \(f\circ (r\circ r)\) to \(T\) (first do \(r\) twice, then do \(f\)) we obtain reflection over the line connecting \(3\) to the midpoint of line segment \(\overline{12}\text{.}\) In fact, every symmetry of \(T\) can be obtained by composing applications of \(f\) and applications of \(r\text{.}\)

For convenience of notation, we omit the composition symbols, writing, for instance, \(fr\) for \(f\circ r\text{,}\) \(r\circ r\) as \(r^2\text{,}\) etc. It turns out there are exactly six symmetries of \(T\text{,}\) namely:

  1. the map \(e\) from \(T\) to \(T\) sending every element to itself;

  2. \(f\) (that is, reflection over the line connecting \(1\) and the midpoint of \(\overline{23}\));

  3. \(r\) (that is, clockwise rotation by \(120^{\circ}\));

  4. \(r^2\) (that is, clockwise rotation by \(240^{\circ}\));

  5. \(fr\) (that is, reflection over the line connecting \(2\) and the midpoint of \(\overline{13}\)); and

  6. \(fr^2\) (that is, reflection over the line connecting \(3\) and the midpoint of \(\overline{12}\)).

Declaring that \(f^0=r^0=e\text{,}\) the set

\begin{equation*} D_3=\{e, f, r, r^2, fr, fr^2\}=\{f^ir^j:i=0,1, j=0,1,2\} \end{equation*}

is the collection of all symmetries of \(T\text{.}\)


Notice that \(rf=fr^2\) and that \(f^2=r^3=e\text{.}\)


Let us look at \(D_3\) another way. Note that each map in \(D_3\) can be uniquely described by how it permutes the vertices \(1,2,3\) of \(T\text{:}\) that is, each map in \(D_3\) can be uniquely identified with a unique element of \(S_3\text{.}\) For instance, \(f\) corresponds to the permutation \((23)\) in \(S_3\text{,}\) while \(fr\) corresponds to the permutation \((13)\text{.}\) In turns out that \(D_3 \simeq S_3\text{,}\) via the following correspondence.

\(e\) \(\mapsto\) \(e\)
\(f\) \(\mapsto\) \((23)\)
\(r\) \(\mapsto\) \((123)\)
\(r^2\) \(\mapsto\) \((132)\)
\(fr\) \(\mapsto\) \((13)\)
\(fr^2\) \(\mapsto\) \((12)\)

The group \(D_3\) is an example of class of groups called dihedral groups.


Let \(n\) be an integer greater than or equal to \(3\text{.}\) We let \(D_n\) be the collection of symmetries of the regular \(n\)-gon. It turns out that \(D_n\) is a group (see below), called the dihedral group of order \(2n\). (Note: Some books and mathematicians instead denote the group of symmetries of the regular \(n\)-gon by \(D_{2n}\)—so, for instance, our \(D_3\text{,}\) above, would instead be called \(D_6\text{.}\) Make sure you are aware of the convention your book or colleague is using.)


Throughout this course, if we are discussing a group \(D_n\) you should assume \(n\in \Z^+\text{,}\) \(n\geq 3\text{,}\) unless otherwise noted.


We say that an element of \(D_n\) is written in standard form if it is written in the form \(f^ir^j\) where \(i\in \{0,1\}\) and \(j\in \{0,1,\ldots,n-1\}\text{.}\)


While \(D_3\) is actually isomorphic to \(S_3\) itself, for \(n>3\) we have that \(D_n\) is not isomorphic to \(S_n\) but is rather isomorphic to a proper subgroup of \(S_n\text{.}\) When \(n>3\) you can see that \(D_n\) cannot be isomorphic to \(S_n\) since \(|D_n|=2n \lt n! = |S_n|\) for \(n>3\text{.}\)

It is important to be able to do computations with specific elements of dihedral groups. We have the following theorem.


  1. Write \(fr^2f\) in \(D_3\) in standard form. Do the same for \(fr^2f\) in \(D_4\text{.}\)

  2. What is the inverse of \(fr^3\) in \(D_5\text{?}\) Write it in standard form.

  3. Explicitly describe an isomorphism from \(D_4\) to a subgroup of \(S_4\text{.}\)


Classify the following groups up to isomorphism. (Hint: You may want to look at the number of group elements that have a specific finite order.)

\begin{equation*} \Z, \Z_6, \Z_2, S_6, \Z_4, \Q, 3\Z, \R, S_2, \R^*, S_3,\Q^*, \C^*, \langle \pi\rangle \text{ in } \R^*, \end{equation*} \begin{equation*} D_6, \langle (134)(25)\rangle \text{ in } S_5, \R^+, D_3, \langle r \rangle \text{ in } D_4, 17\Z \end{equation*}