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QUESTION

State Burnside's Formula, carefully defining the terms used in the
formula.

Describe the elements of the rotation group of the cube, giving
the order of each element, its fixed set, and describing the
orbits of the faces for each rotation.

Use this to find the number of distinct ways there are to label
the faces of a cube with five colours, where each colour may be
used more than once. (As usual, \lq\lq distinct'' means that the
labellings can be distinguished up to a rotation of the cube, so
you will need to consider the action of the rotation group of the
cube on the set of all possible labellings.)




ANSWER


Burnsides formual:

$$r|G|=\sum_{g\in G}|X_g|$$

where $G$ acts on a set $X$, $X_g=\{x\in X|gx=x\}$ and $r$= number
of orbits.

Description

\begin{tabular}{c|l|ccc}
type&transformation&number&order&face orbits\\ \hline
A&identity&1&1&6\\ \hline B&rotations of&8&3&2\\ &order 3\\ &about
diagonal\\ \hline C&rotations of order&6&2&3\\ &2 about line\\
&bisecting 2 opposite\\ &edges\\ \hline D&rotations about&3&2&4\\
&lines joining\\ &midpoints&6&4&3\\ &of opposite faces\\
\end{tabular}

Number of distinguishable dice=number of orbits of rotation group
on all colourings.

By Burnside's formula

\begin{eqnarray*}
24r&=&5^6+8.5^2+6.5^3+3.5^4+6.5^3\\ &=&19200
\end{eqnarray*}

so $r=\frac{19200}{24}=800$.




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