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Let $C$ be a set of colors. 
A {\it coloring} of $X$ is a function ${\bf c}: X \rightarrow C$

Let ${\cal C}$ be a set of colorings of $X$.

Example:  If $X = \{1, 2, 3\}$,  $C = \{$red, blue$\}$, then let \hb
${\cal C} = \{  {\bf c_1},  {\bf c_2},  {\bf c_3},  {\bf c_4},  {\bf c_5},  {\bf c_6},  {\bf c_7},  {\bf c_8} \}$ where \hb
${\bf c_i}:    \{1, 2, 3\}  \rightarrow \{$red, blue$\}$  $\forall i$ and
\s
 ${\bf c_1}(j) =$ blue for all $j$;
\s
${\bf c_2}(1) =$ blue, ${\bf c_2}(2)$ = blue , ${\bf c_2}(3)$ = red;
\s

${\bf c_3}(1) =$ blue, ${\bf c_2}(2)$ = red, ${\bf c_2}(3)$ = blue;
\s
${\bf c_4}(1) =$ red, ${\bf c_2}(2)$ = blue, ${\bf c_2}(3)$ = blue;
\s
${\bf c_5}(1) =$ blue, ${\bf c_2}(2)$ = red, ${\bf c_2}(3)$ = red;
\s
${\bf c_6}(1) =$ red, ${\bf c_2}(2)$ = blue, ${\bf c_2}(3)$ = red;
\s
${\bf c_7}(1) =$ red, ${\bf c_2}(2)$ = red, ${\bf c_2}(3)$ = blue;
\s
${\bf c_8}(j) =$ red for all $j$.



Let $G$ be a set of permutations.

A permutation $f$ acts on a coloring ${\bf c}$ as follows:  


$(f *  {\bf c})(x) = ({\bf c}  \circ  f^{-1})(x) = {\bf c}(f^{-1}(x))$ 

Ex:  Suppose $f$ is the permutation 231.  Then 

$f *  {\bf c_2}(1) = {\bf c_2}(f^{-1}(1)) = {\bf c_2}(3)$ =  red.
 \s
$f *  {\bf c_2}(2) = {\bf c_2}(f^{-1}(2)) = {\bf c_2}(1)$ =  blue.
\s
$f *  {\bf c_2}(3) = {\bf c_2}(f^{-1}(3)) = {\bf c_2}(2)$ =  blue.


Thus $f *  {\bf c_2} = {\bf c_4}$



Defn:  Let $G$ be a set of permutations.
  ${\bf c_1} \sim  {\bf c_2}$ if there exists an $f \in G$ such that $f *  {\bf c_1} = {\bf c_2}$

Note $\sim$ is an equivalence relation.


14.2:  Burnside's Theorem.


Let $G({\bf c}) = \{f \in G ~|~ f*{\bf c} = {\bf c}\}$. \hfil 

Let ${\cal C}(f) = \{ {\bf c} \in {\cal C} ~|~ f*{\bf c} = {\bf c}\}$.


Thm 14.2.1a:  $G({\bf c})$ is a group.


Thm 14.2.1b:  $g*{\bf c} = f*{\bf c}$ if and only if $f^{-1} \circ g \in G({\bf c})$.

Thm 14.2.2:  $|\{f* {\bf c} ~|~ f \in G \}| = {|G| \over |G({\bf c})|}$

Note  $|\{f* {\bf c} ~|~ f \in G \}| $ = the number of different colorings which are equivalent to  ${\bf c}$.




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