The Dihedral Group of a Bracelet

A Dihedral group, \(D_n\), is the group of rotations and reflections (the symmetries) of a regular \(n\)-gon. The order (size) of the group is \(2n\).   We considered the rotations in the Cyclic group of a necklace  and will consider here the addition of reflectional symmetries in order to count the number of different bracelets that can be made from \(n\) beads of \(a\) different colours.

We continue our count using a set of six beads.

The Cayley graph of \(D_6\) is :

OldTrout, Going Postal

The reflectional symmetries of a regular hexagon are :

OldTrout, Going Postal

There are three lines of reflection through opposite vertices and three lines of reflection through opposite midpoints.

In order to be able to count the number of different bracelets, we need to know the cycle permutations of these six reflections.

The cycle permutations through the vertices :

OldTrout, Going Postal

The cycle permutations through the mid-points :

OldTrout, Going Postal

We see by inspection that the three reflections through the vertices give two fixed points and two cycles of length two and the other three reflections through the mid-points give three cycles of length two.

We use this information in the cycle index polynomial for a Dihedral group.

If \(n\) is even, then

\(Z(D_n) = \frac12 Z(C_n) + \frac14 \bigl (a_1^2a_2^{(n-2)/2} + a_2^{n/2} \bigr)\),

where \(\displaystyle Z(C_n) = \frac 1n \sum_{d|n} \varphi (d) a_d^{n/d} \)

When \(n = 6\)

\(Z(D_6) = \frac 12 Z(C_6) + \frac 14 \bigl (a_1^2a_2^2 + a_2^3 \bigr) \)

When \(a = 1\) the sum is trivial,

\(Z(D_6) = \frac 12 Z(C_6) + \frac 14 (1^2 \cdot 1^2 + 1^3) = \frac 12 + \frac 12 = 1 \)

When \(a = 2\),

\(Z(D_6) = \frac 12 Z(C_6) + \frac 14 (2^2\cdot2^2 + 2^3) = 7 + \frac 14 (16 + 8) = 7 + 6 = 13 \)

When \(a = 3\),

\(Z(D_6) = \frac 12 Z(C_6) + \frac 14 (3^2\cdot3^2 + 3^3) = 65 + \frac 14 (81 + 27) = 65 + 27 = 92 \)

 

Notes :

If \(n\) is odd, then

\(Z(D_n) = \frac 12 Z(C_n) + \frac 12 a_1a_2^{(n-1)/2}\)  since all the axes of reflection are through a vertex and the opposite mid-point.

For \(\varphi(n)\) see Euler’s Totient Function.

For \(Z(C_n)\) see the Cyclic Group of a Necklace.

Dihedral groups \(n \gt 2\)  are non-Abelian.  They do not commute :

\(rs \not =  sr\)

\(sr = r^{-1}s = r^5s\)

\(D_6 = \{r^ms^n : m = 0, …, 5, n = 0, 1 ; r^6 = s^2 = e, sr =r^5s \}\),

where \(r\) is a rotation and \(s\) is a reflection and they are the generators of the group.

This gives the elements of \(D_6\) as

\(e = r^0, r, r^2, r^3, r^4, r^5, s, rs, r^2s, r^3s, r^4s, r^5s\).

 

© OldTrout \(2018\)
 

No Audio file – does not translate well