Affine Distance‐Transitive Groups

A fruitful way of studying groups is by means of their actions on graphs. A very special and interesting class of groups are the ones acting distance-transitively on some graph. A group G acting on a connected graph F = (VT, ET) is said to act distance-transitively if its action on each of the sets {(x, y)\ x, y e VT, d(x, y) = i} is transitive. Here VT and ET stand for the vertices and edges of T respectively, d stands for the usual distance in T, and i runs through {0, 1, 2, ..., diam(F)}. A graph is called a distance-transitive graph if it admits a group acting distancetransitively on it. Moreover, in this case the group is called a distance-transitive group. In general there is no hope of classifying all distance-transitive graphs, as observed by Cameron (cf. §7.8 of Brouwer, Cohen, and Neumaier [8]). However, if there are only finitely many vertices, then a complete classification is in sight. First, if the group G acts imprimitively on a distance-transitive graph T, then by a result of Smith [31], there is a natural way to construct a new graph from T that admits a distance-transitive group acting primitively on it. In the classification project we usually restrict ourselves to the primitive groups. For many of the known primitive distance-transitive graphs it is known whether they have an imprimitive parent or not (see Hemmeter [18] and van Bon and Brouwer [6]). Next notice that when the diameter is equal to 1, the graph is just a clique, and G acts distance-transitively on it if and only if G is a 2-transitive group. These groups are known (cf. Cameron [10], Cohen and Zantema [11], Hering [19], Kan tor [21], Liebeck [24], Curtis, Kantor and Seitz [13] and many others). Their classification depends on the classification of finite simple groups. When the valency of the graph equals 2, that is, each vertex is adjacent to precisely two others, the graph is a polygon and G must be a dihedral group. In the case where G is a rank-3 permutation group, there are two distance-transitive graphs associated with it, one being the complement of the other. This case has been completely settled by Foulser [15], Foulser and Kallaher [16], Kantor and Liebler [22], Liebeck [25] and Liebeck and Saxl [28]. For the general primitive case, the first determination of the structure of G was made by Praeger, Saxl and Yokoyama [29] in 1987. Their theorem, based on the O'Nan Scott Theorem [9, 26, 30], states that if G acts primitively and distance-transitively on a finite graph F of diameter at least 2 and valency at least 3, then either F is a Hamming graph and G is a wreath product, or G is almost simple or affine. By the classification of finite simple groups and the knowledge of their automorphism groups and maximal subgroups, a full classification of groups and graphs in the