Automorphisms of Trivalent Graphs

In [7] and [8], Tutte considered a vertex-transitive group of automorphisms of a finite, connected, trivalent graph. He showed that if the stabilizer of a vertex is transitive on adjacent vertices, then its order divides 3 . 24. As observed by Sims [5], the hypothesis is equivalent to the following group-theoretic conditions: a) G is a finite group generated by a pair of subgroups {P1, Pd, b) i Pi: Pi n P2 1= 3 for i = 1, 2, c) no non-trivial normal subgroup of G is contained in P1 A, d) P1 and P2 are G-conjugate. What happens if we drop condition d) or, what is essentially the same thing, replace "vertex transitive" by "edge transitive"? This question is primarily motivated by the examples afforded by the rank 2 BN pairs over GF(2). In this case, the trivalent graph mentioned above is the so-called "building" associated to the BN pair [6]. In this paper, we classify all pairs of subgroups (P1, P2) for which hypotheses a), b) and c) are satisfied. There are precisely fifteen such pairs, and in particular, we find that P1 n P2 has order dividing 27. In order to describe the results more completely, let us define an amalgam to be a pair of group monomorphisms (5, 02) with the same domain: P, 1 B P2. We will say that (01, 02) is finite if both co-domains P1, P2 are finite. In this case, we define the index of the amalgam to be the pair of indices (I Pl: im s1 I, P2: im 02 1). By a completion of the amalgam we mean a pair of homomorphisms (*1, '2) to some group G making the obvious diagram commute, i.e., such that 01* = 102'2 (in right-hand notation). By abuse of notation, we may say that G is a completion of the amalgam. Of course, we always have the trivial completion g1 = g2 = 1We also always have the universal completion, usually known as the amalgamated product, from