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\title{Colored Symmetry}
\author{Tobin Fricke}
\date{Spring 2002}

Foo 


The \emph{stabilizer subgroup} of an element $x \in X$ is the subgroup of $G$ containing exactly those elements which preserve $x$.  We write $G_{x} = \{g \in G : gx = x\}$.

The \emph{orbit} of a point $x \in X$ is defined as the subset of $X$ reachable from $x$ by the actions of $G$.   We write $G(x) = \{ gx : g \in G \}$.  Orbits may be used to form an equivalence relation on X, where two points $x, y \in X$ are considered equivalent if they lie in the same orbit; that is, if $y = gx$ for some $g \in G$.

Let $H = G_e$ where $e$ is the identity element of $G$.  

A \emph{generic orbit} $G(x)$ has the property that it stabilizer group $G_x$ is trivial, and that $|G(x)| = |G|$.  The points in a generic orbit may thus be put into one-to-one correspondence with elements of $G$.  Existence??


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