Three-Colorings of Finite Groups or an Algebra of Nonequalities

Abstract

Graph coloring theory is a remarkable branch of mathematics. It begins with one of the simplest of ideas, that of coloring a geographical map or the vertices of a graph in such a way that adjacent regions or vertices are colored differently. And yet, as the celebrated four color and Heawood map-coloring theorems show, the existence of such colorings reflects profoundly the fundamental topological properties of the graph. In general terms, one may say that the set of colorings of a graph contains structural information about the graph. This paper, going by the principle that a good way to understand a theory is to try to construct a parallel one, will develop a coloring theory in a quite different context, that of groups. The ultimate conclusion will be the same as it was for graphs: The set of colorings contains structural information about the group. Along the way to this conclusion, there will be a chance to think about what makes graph colorings so special, and also, we hope, a pleasant jaunt through elementary group theory, seen from a novel perspective.' What is the unique essential ingredient in the idea of coloring a graph? Perhaps it is the fact that making one choice when defining a coloring does not dictate the next step, but merely limits its possibilities. Coloring point 1 in FIGURE 1 red does not force point 2 to be blue, but only constrains it not to be red. Algebraic constructions, in contrast, are usually much more deterministic-once the value of a homomorphism is defined at x, it is fixed forever at all xn . There is a heady freedom in defining colorings, like dealing with a card sharper or a gambit player who smiles and says Pick any card! or Now make any move at all! And yet the freedom is often only temporary. Some point in FIGURE 1 eventually must be assigned a third color. There is, similarly, no 3-coloring at all of FIGURE 2. In colorings, as in cards or chess, a few moves later one may be locked in-or out-just as tightly as with the homomorphism! An n-coloring of a graph is thus a mapping into a set of n colors, assigned with absolute freedom except for one special kind of constraint: If two points are directly related by the graph structure (that is, are adjacent), then the simplest choice (coloring them both the same) is forbidden. What would be a parallel construction for