George Pólya, who made great contributions not only to mathematics itself, but also to the understanding of how and why we do mathematics - or perhaps one should say 'how and why we should do mathematics' - was particularly fascinated by the Platonic solids and first introduced his notion of homologues in connection with the study of their symmetry; they later played an important role in one of his most important contributions to the branch of mathematics know as combinatorics, namely, the Pólya Enumeration Theorem (see [P1] for an intuitive account). Let us describe this notion of homologues in terms of symmetry groups. We believe that we are thereby increasing the scope of the notion and entirely maintaining the spirit.

Let A be a geometrical configuration with symmetry group G_A, and let B be a subset of A. Thus, for example, A may be a polyhedron and B a face of that polyhedron. We consider the subgroup G_AB of G_A consisting of those motions in the symmetry group G_A of A which map B to itself. Now subgroups partition a group into cosets: If K is a subgroup of H, we define a (right) coset of K in H as a collection of elements kh, with h fixed and k varying over K. We call h a representative of this coset, which we write Kh. Any two cosets Kh, Kh¢ are either disjoint or identical (this is easy to prove), so we may imagine that we have picked a set of coset representatives, one for each coset. In the case in which we are interested the group H is finite so we may write, for some m

where it is understood that the union is disjoint. Notice that m, which appears in (4.1) and which we call the index of K in H, is just the ratio of the order of H to the order of K. An example was provided earlier with H = S_n, the symmetric group, and K = A_n, the alternating group. Then m = 2.

Reverting to our geometrical situation , we consider a coset of G_AB in G_A, that is, a set G_ABg, g Î G_A. Every element in G_ABg sends B to the same subset Bg of A. The collection of these subsets is what Pólya called the collection of homologues of B in A. We see that the set of homologues of B is in one-one correspondence with the set of cosets of G_AB in G_A.

Example 4.1: Consider the pentagonal dipyramid A of Figure 5(b) of [Rec]. We may specify any motion in the symmetry group of A by the resulting permutation of its vertices 1,2,3,4,5,6,7. In fact, G_A is the dihedral group D₅, with 10 elements, given by the following permutations sending (1 2 3 4 5 6 7), resepectively, to