A relation $\sim$ on $C$ is equivalence if it is:
- reflexive: $c\sim c$ for all colorings $c$ belonging to $C$
- symmetric: $c_1\sim c_2$ implies $c_2\sim c_1$
- transitive: $c_1\sim c_2$ and $c_2\sim c_3$ implies $c_1\sim c_3$
A set of objects $S$, a set of colorings of these objects $C$ and a group of permutations $G$ representing symmetries possessed by configurations of the objects. Two colorings in $C$ are equivalent if there is a permutation in $G$ that transforms a coloring into the other.
Show that $\sim$ is an equivalence relation on $C$. How do you prove that?
As well as that, prove that $G_c$ is a subgroup of $G$.