  We introduce coloring groups, which are permutation groups obtained from a
proper edge coloring of a graph. These groups generalize the generalized toggle
groups of Striker (which themselves generalize the toggle groups introduced by
Cameron and Fon-der-Flaass). We present some general results connecting the
structure of a coloring group to the structure of its graph coloring, providing
graph-theoretic characterizations of the centralizer and primitivity of a
coloring group. We apply these results particularly to generalized toggle
groups arising from trees as well as coloring groups arising from the
independence posets introduced by Thomas and Williams.
