Semisymmetric cubic graphs of twice odd order

Suppose that Γ is a connected graph and G is a subgroup of the automorphism group Aut(Γ) of Γ. Then Γ is G-symmetric if G acts transitively on the arcs (and so the vertices) of Γ and Γ is G-semisymmetric if G acts edge transitively but not vertex transitively on Γ. If Γ is Aut(Γ)symmetric, respectively, Aut(Γ)-semisymmetric, then we say that Γ is symmetric, respectively, semisymmetric. If Γ is G-semisymmetric, then we say that G acts semisymmetrically on Γ. If Γ is G-semisymmetric, then the orbits of G on the vertices of Γ are the two parts of a bipartition of Γ. Semisymmetric cubic graphs (graphs in which every vertex has degree 3) have been the focus of a number of recent articles, we mention specifically [9, 11, 10, 30, 31, 32, 33, 34] where infinite families of such graphs are presented and where the semisymmetric graphs of order 2pq, 2p, 6p with p and q odd primes are determined (with the help of the classification of the finite simple groups). We also remark that a catalogue of all the semisymmetric cubic graphs of order at most 768 has recently been obtained by Conder et. al. [7]. The objective of this article is to partially describe all groups which act semisymmetrically on a cubic graph of order twice an odd number. Thus our result reduces some of the aforementioned investigations to checking group orders in our list. We also mention that our theorems call upon only a small number of characterization theorems used in the classification of the finite simple groups and not on the whole classification itself. Suppose that Γ is a G-semisymmetric cubic graph. Let {u, v} be an edge in Γ. Set Gu = StabG(u), Gv = StabG(v) and Guv = Gu ∩ Gv. Then, as G acts edge transitively on Γ and u is not in the same G-orbit as v, we have [Gu : Guv] = [Gv : Guv] = 3. Suppose that K ⊳G and K ≤ Guv. Then K fixes every edge of Γ and hence K = 1. As Γ is connected, the subgroup 〈Gu, Gv〉 acts transitively on the edges of Γ and so we infer that G = 〈Gu, Gv〉. We have shown that G satisfies • G = 〈Gu, Gv〉; • [Gu : Gu ∩Gv] = [Gv : Gu ∩Gv] = 3; and