A Note on Groups Associated with 4-arc-transitive Cubic Graphs

A cubic (trivalent) graph F is said to be 4-arc-transitive if its automorphism group acts transitively on the 4-arcs of r (where a 4-arc is a sequence «;0, vv...,vi of vertices of F such that t;,_j is adjacent to vt for 1 ^ I < 4, and vt-1 ^ vi+1 for 1 < i < 4). In his investigations into graphs of this sort, Biggs defined a family of groups 4(a), for m = 3 ,4 ,5 . . . , each presented in terms of generators and relations under the additional assumption that the vertices of a circuit of length m are cyclically permuted by some automorphism. In this paper it is shown that whenever m is a proper multiple of 6, the group 4(a) is infinite. The proof is obtained by constructing transitive permutation representations of arbitrarily large degree.