A NOTE ON GROUPS ASSOCIATED WITH 4-ARC-TRANSITIVE CUBIC GRAPHS

A cubic (trivalent) graph-GAMMA is said to be 4-arc-transitive if its automorphism group acts transitively on the 4-arcs of GAMMA (where a 4-arc is a sequence upsilon-0, upsilon-1,..., upsilon-4 of vertices of GAMMA such that upsilon-i-1 is adjacent to upsilon-i for 1 less-than-or-equal-to i less-than-or-equal-to 4, and upsilon-i-1 not-equal upsilon-i+1 for 1 less-than-or-equal-to i < 4). In his investigations into graphs of this sort, Biggs defined a family of groups 4+(a(m)), for m = 3,4,5..., each presented in terms of 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(m)) is infinite. The proof is obtained by constructing transitive permutation representations of arbitrarily large degree.