A Note on Groups Associated with 4-Arc-Transitive Cubic Graphs

A cubic (trivalent) graph is said to be 4-arc-transitive ifits automorphism group acts transitively on the 4-arcs of (wherea 4-arc is a sequence v₀, v₁, ... v₄ of vertices of such thatv_i–1 is adjacent to v_i for 1 i 4, and v_i–1 v_i+1for 1 i < 4). In his investigations into graphs of thissort, Biggs defined a family of groups 4⁺(a^m), for m = 3,4,5...,each presented in terms of generators and relations under theadditional assumption that the vertices of a circuit of lengthm are cyclically permuted by some automorphism. In this paperit is shown that whenever m is a proper multiple of 6, the group4⁺(a^m) is infinite. The proof is obtained by constructing transitivepermutation representations of arbitrarily large degree.