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Finite vertex-biprimitive edge-transitive tetravalent graphs✩ Cai Heng Li a , Hua Zhang b,a,∗ a

School of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia

b

School of Mathematics, Yunnan Normal University, Kunming 650091, PR China

article

info

Article history: Received 21 December 2012 Received in revised form 29 October 2013 Accepted 9 November 2013 Available online 23 November 2013

abstract Ivanov and Iofinova classified vertex-biprimitive edge-transitive cubic graphs in 1985. As a natural generalization of Ivanov and Iofinova’s work, in this paper we present a classification of tetravalent graphs which are G-vertex-biprimitive and G-edge-transitive for some automorphism group G of the graphs. © 2013 Elsevier B.V. All rights reserved.

Keywords: Algebraic graph theory Edge-transitive graphs Vertex-biprimitive

1. Introduction In a highly cited article [8], Ivanov and Iofinova classified vertex-biprimitive edge-transitive cubic graphs, based on the classification of amalgams of edge-transitive cubic graphs obtained by Goldschmidt [7]. As a natural generalization of Ivanov and Iofinova’s work, in this paper we classify vertex-biprimitive edge-transitive tetravalent graphs. Recall that a bipartite graph Γ = (V , E ) with two parts U and W is called G-biprimitive for some G ≤ Aut Γ if G acts primitively on both U and W , and is called G-edge-transitive if G is transitive on the edge set E. An arc of a graph is an ordered pair of adjacent vertices, and a graph Γ is called G-arc-transitive if G ≤ Aut Γ is transitive on the set of arcs. The result of this paper is the following theorem. Theorem 1.1. Let Γ be a finite connected bipartite graph of valency 4. Assume that Γ is G-biprimitive and G-edge-transitive, where G ≤ Aut Γ . Assume further that G is intransitive on V . Then for an edge {α, β}, one of the following holds: (i) Γ = K4,4 , and Z22 : Z3 ≤ G < Aut Γ = S4 ≀ Z2 . (ii) |V | = 2p, 2p2 , or 2p3 , with p a prime, and one of the following holds, respectively, G = Zp : Z4 with p > 5 and 4 | (p − 1), and Aut Γ = G × Z2 . G = Z2p : Z4 with 4 - (p − 1), or Z2p : D8 , and Aut Γ = (Z2p : D8 ) × Z2 .

G = Z3p : A4 or Z3p : S4 , and Aut Γ = (Z3p : S4 ) × Z2 . (iii) Γ is the standard double cover of a vertex-primitive arc-transitive tetravalent graph. (iv) G = PSL2 (p), where p is a prime with p ≡ ±13, ±37 (mod 40), Gα ∼ = Gβ ∼ = A4 , and Γ is one of graphs, where ε = 1 or −1 such that 3 divides p + ε . (v) G = PSL2 (3f ) with f ≥ 3 prime, Gα ∼ = Gβ ∼ = A4 , and Γ is one of

3f −1 −1 2

p+ε 6

non-isomorphic

non-isomorphic graphs.

✩ This work is partially supported by an ARC Discovery Project Grant, and the second author is partially supported by NNSF (No. 1116058) and YNSF (No. 2011FZ087). ∗ Corresponding author at: School of Mathematics, Yunnan Normal University, Kunming 650091, PR China. E-mail addresses: [email protected] (C.H. Li), [email protected], [email protected] (H. Zhang).

0012-365X/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.disc.2013.11.006

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C.H. Li, H. Zhang / Discrete Mathematics 317 (2014) 33–43 Table A Amalgams associated with almost simple groups. K

Kα

Kβ

Aut Γ

PGL2 (11) PSL2 (13) PSL2 (23) PSL2 (p) PSL3 (3) PSL3 (3) PGL3 (7) PGU3 (5) PSp4 (3) G2 (3) M12 Th

D24 D12 D24 S4 S4 32 : 2S4 32 : 2A4 32 : 2A4 [33 ] : S4 (31++2 × 32 ) : 2S4 32 : 2S4 [39 ].2S4

S4 A4 S4 S4 S4 32 : 2S4 62 : S3 62 : S3 [33 ] : 2A4 (31++2 × 32 ) : 2S4 32 : 2S4 32 .[37 ].2S4

PGL2 (11) PGL2 (13) PSL2 (23) PGL2 (p) PSL3 (3).2 PSL3 (3).2 PGL3 (7).2 PGU3 (5).2 PSp4 (3).2 G2 (3).2 M12 .2 Th

Comments

p ≡ ±1 (mod 8) Projective plane

Generalized hexagon Weiss graph Discovered in [3]

(vi) G is an almost simple group, there exists a normal subgroup K ▹ G such that K , Kα , Kβ and Aut Γ lie in Table A, and for each case there is a unique graph. Moreover, if Γ is not arc-transitive, then G is one of the following groups: PSL2 (p) as in part (iv), PSL2 (3f ) as in part (v), or PGL2 (11), PSL2 (13), PSL2 (23), PGL3 (7), PGU3 (5), PSp4 (3), or Th, as in Table A. Remarks on Theorem 1.1: (1) Unlike the cubic case, amalgams for tetravalent graphs are not known yet. The proof of Theorem 1.1 depends on the classification of primitive permutation groups which have soluble stabilizers, as given in [10]. (2) Finite vertex-primitive arc-transitive tetravalent graphs are classified in [9], and so graphs in part (iii) of Theorem 1.1 are known. (3) All the graphs in (iv) are the standard double covers of certain undirected graphs or digraphs. Some of the graphs in (iv) are the standard double covers of graphs in (iii), while others are not, depending on the choice of Gβ (see Lemma 5.7 for details). The graphs in (v) are the standard double covers of certain digraphs. (4) To our best knowledge, the PGL3 (7)-graph and PGU3 (5)-graph are new examples, whereas all other graphs appearing in Theorem 1.1 have been known. (5) Some graphs in Theorem 1.1 are actually vertex-transitive, such as some of the graphs in part (iii), and the graphs associated with the groups and stabilizers of the fourth row and the fifth row in Table A. The reader should keep in mind that by ‘‘vertex-biprimitive’’ we mean Aut Γ -vertex-intransitive. In this paper we only present G-vertex-biprimitive graphs because in some cases (for example in part (iv)) the vertex transitivity depends on the choice of Gβ . The layout of this paper is as follows. In Section 2 we present some of the graphs appearing in Theorem 1.1. In Section 3 we give some properties about the vertex stabilizers of edge-transitive tetravalent graphs. Then in Section 4, we give a reduction of the proof of Theorem 1.1 to the case of almost simple groups. Finally in Section 5, we deal with the groups of almost simple type, and in Section 6, we complete the proof of the main theorem. 2. Examples In this section, we construct and study the graphs appearing in Theorem 1.1. Let Γ = (V , E ) be a connected bipartite G-edge-transitive regular graph, say of valency k, and let {α, β} be an edge. Then the edge stabilizer Gαβ coincides with Gα ∩ Gβ , Gα is transitive on Γ (α) with stabilizer (Gα )β = Gαβ , and Gβ is transitive on Γ (β) with stabilizer (Gβ )α = Gαβ . Thus |Gα : Gαβ | = |Gβ : Gαβ | = k. Since Γ is connected, it follows that ⟨Gα , Gβ ⟩ = G. Conversely, for a group G and subgroups L, R < G such that L ∩ R is core-free, one can define a G-edge-transitive graph with vertex set V = [G : L] ∪ [G : R], the disjoint union of the sets of the right cosets of L in G and R in G, such that Lx ∼ Ry ⇐⇒ Ry ∼ Lx ⇐⇒ xy−1 ∈ LR. This graph is called a coset graph, and is denoted by Cos(G, L, R). This coset graph is G-vertex-intransitive, and is not necessarily regular, the valencies of which are |L : L ∩ R| and |R : L ∩ R|. A detailed study of such graphs can be found in [6]. First we list the following three results for later use, the proof of Lemma 2.1 is simple and can be seen in [6], while Lemma 2.3 is clear. Lemma 2.1. Let Γ = (V , E ) be a graph and let G ≤ Aut Γ be transitive on E and intransitive on V . Then for an edge {α, β}, Γ is isomorphic to Cos(G, Gα , Gβ ). g

g

Lemma 2.2. The coset graphs Cos(G, L1 , R1 ) = Cos(G, L2 , R2 ) if and only if L2 = L1 and R2 = R1 for some g ∈ G.

C.H. Li, H. Zhang / Discrete Mathematics 317 (2014) 33–43

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Proof. Let α1 , β1 be the vertices corresponding to L1 , R1 , respectively. Then the graph Cos(G, L1 , R1 ) has vertex set [G : L1 ] ∪ [G : R1 ] and edge set (α1 , β1 )G . Let (α2 , β2 ) be another edge, where α2 ∈ [G : L1 ] and β2 ∈ [G : R1 ]. Then the edge set of Cos(G, L1 , R1 ) also has the form (α2 , β2 )G . g g Let g ∈ G be such that (α1 , β1 )g = (α2 , β2 ), and let L2 = Gα2 and R2 = Gβ2 . Then L1 = L2 and R1 = R2 . Thus Cos(G, L1 , R1 ) can also be expressed as Cos(G, L2 , R2 ), namely Cos(G, L1 , R1 ) = Cos(G, L2 , R2 ). Conversely, suppose that Γ := Cos(G, L1 , R1 ) = Cos(G, L2 , R2 ). Let αi , βi be the vertices corresponding to Li , Ri , respectively, where i = 1, 2. Then (α1 , β1 ) and (α2 , β2 ) are two edges. Hence there exists g ∈ G such that (α1 , β1 )g = (α2 , β2 ). So Lg1 = L2 and Rg1 = R2 . Lemma 2.3. For a group G and subgroups L, R < G such that ⟨L, R⟩ = G, L ∩ R is core-free, and |L : L ∩ R| = |R : L ∩ R| = 4, the coset graph Cos(G, L, R) is connected and G-edge-transitive of valency 4. If, in addition, both L and R are maximal subgroups of G, then the graph is G-vertex-biprimitive. There is another kind of coset graphs which can be defined as follows. Let G be a finite group, and let H be a core-free subgroup of G. Denote by [G : H ] the set of right cosets of H in G. For an element g ∈ G, the coset graph of G with respect to H and g is the digraph with vertex set [G : H ], such that Hx and Hy are adjacent if and only if yx−1 ∈ HgH. This coset graph is G-vertex-transitive, and is denoted by Cos(G, H , HgH ). For a (directed or undirected) graph Γ with vertex set V Γ and arc set AΓ , the standard double cover of Γ is defined to be the undirected bipartite graph Γ˜ with vertex set V Γ × {1, 2}, and two vertices (u, 1) and (v, 2) are adjacent if and only if (u, v) ∈ AΓ . The two parts of the graph Γ˜ are V Γ × {i}, where i = 1, 2. The above two coset graph constructions are related in the following way. Lemma 2.4. Let G be a group and L a subgroup of G. Assume that g ∈ G is such that L ∩ Lg is core-free. Then Γ = Cos(G, L, Lg ) is the standard double cover of Cos(G, L, LgL), and Γ is arc-transitive if there exists x ∈ G such that x2 ∈ NG (L), and Lx = Lg . Proof. By [6, Lemma 3.8], Γ = Cos(G, L, Lg ) is the standard double cover of Cos(G, L, LgL). If there exists x ∈ G such that x2 ∈ NG (L) and Lx = Lg , then Γ = Cos(G, L, Lg ) = Cos(G, L, Lx ). Let α, β be the vertices of Γ corresponding to L and Lg , respectively. Then (α, β) is an arc, and as x2 ∈ NG (L), x induces an automorphism of Γ that interchanges α and β . So Cos(G, L, Lx ) is arc-transitive. The following examples are associated with certain almost simple groups. First we look at a few well-known graphs. 1+2 Example 2.5. Let G be one of the groups: PSL3 (3), G2 (3) or M12 , and let H < G be isomorphic to 32 : 2S4 , (3+ × 32 ) : 2S4 or 2 3 : 2S4 , respectively. Then the subgroups of G which are isomorphic to H are maximal and form two conjugacy classes. Let L, R < G be isomorphic to H and non-conjugate. Then ⟨L, R⟩ = G, and both L and R contain Sylow 3-subgroups of G, and hence up to conjugacy we may assume that L ∩ R contains a Sylow 3-subgroup, say Q . First we consider the case G = PSL3 (3). By the ATLAS [1] and the maximality of L, the normalizer of Q in G is contained in a maximal subgroup of G which is isomorphic to L. By the structure of L, the normalizer is isomorphic to 32 : 2S3 . Thus we have 32 : 2S3 = NL (Q ) = NG (Q ) ≥ NR (Q ) = 32 : 2S3 , so L ∩ R ≥ 32 : 2S3 , and |L : L ∩ R| ≤ 4. Note that |L : L ∩ R| = 2 is clearly impossible, we conclude that |L : L ∩ R| = |R : L ∩ R| = 4. The same is true for the remaining two cases. Therefore the graph Γ = Cos(G, L, R) is connected, G-vertex-biprimitive, and of valency 4. The PSL3 (3)-graph is the incidence graph of the projective plane PG(2, 3), the G2 (3)-graph is a generalized hexagon of order (3, 3), and the M12 -graph was first constructed by Weiss [14].

Example 2.6. Let G be one of the groups: PGL2 (11), PGL2 (13), or PSL2 (23). Then G has two maximal subgroups L ∼ = S4 and R∼ = D24 . Notice that any Sylow 3-subgroup of G is isomorphic to Z3 , so we may assume that both L and R contain a Sylow 3-subgroup Q . Then NL (Q ) = S3 , and NR (Q ) = R. Thus L ∩ R = NL (Q ) = S3 , and |L : L ∩ R| = |R : L ∩ R| = 4. Therefore the graph Γ = Cos(G, L, R) is G-vertex-biprimitive, G-edge-transitive, and of valency 4. For G = PGL2 (13), let T = PSL2 (13). Then T ∩ L = A4 , and T ∩ R = D12 . As T ∩ L and T ∩ R are maximal subgroups of T , the graph Γ is also T -vertex-biprimitive. Example 2.7. The group G = PSp4 (3).2 has two maximal subgroups L = [33 ] : 2S4 and R = [33 ] : (S4 × 2). Both L and R contain Sylow 3-subgroups of G. Thus up to conjugacy, we may choose L, R such that L ∩ R contains a Sylow 3-subgroup, say Q . Then for the same reason as in Example 2.5, we have L∩R = NL (Q ) = NR (Q ), and hence |L : L∩R| = |R : L∩R| = 4. Therefore, the coset graph Γ = Cos(G, L, R) is G-vertex-biprimitive of valency 4. Let T = soc(G) = PSp4 (3). Then T ∩ L = [33 ] : 2A4 , and T ∩ R = [33 ] : S4 . As T ∩ L and T ∩ R are maximal subgroups of T , the graph Γ is also T -vertex-biprimitive. Example 2.8. The group G = PSL3 (3) contains maximal subgroups that are isomorphic to S4 and all of them are conjugate. Hence G has a primitive action of degree 234. Take two copies of this action on a set Ω1 and Ω2 . Then it can be checked with Magma that there exist ω1 ∈ Ω1 and ω2 ∈ Ω2 with Γ = (ω1 , ω2 )G having size 4 × 234, and with the corresponding graph being connected. Thus the graph Γ is G-vertex-biprimitive of valency 4. Moreover, Magma says that the automorphism group of this graph is PΓ L(3; 3). In particular, Γ cannot be a double cover of a vertex-primitive arc-transitive tetravalent graph because Aut(Γ ) has no central element of order 2 (This graph was missed by the authors, we thank the anonymous referee for pointing out this to us.)

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C.H. Li, H. Zhang / Discrete Mathematics 317 (2014) 33–43

The next example presents a few infinite families of graphs. Example 2.9. Let T = PSL2 (q), where q = pf with p prime. (1) Let q = p ≡ ±13, ±37 (mod 40), or q = 3f with f ≥ 3 prime. Then in both cases the group T contains a maximal subgroup H = A4 . Let h ∈ H be of order 3. For the former case, we have CT (h) = Z p+1 or Z p−1 . Let g ∈ CT (h) \ ⟨h⟩. For 2

2

the latter case, ⟨h⟩ is a proper subgroup of a Sylow 3-subgroup, say Q , of T = PSL2 (3f ), and Q is abelian. Let g ∈ Q \ ⟨h⟩. Then in each case, the graph Cos(T , H , H g ) is T -vertex-biprimitive of valency 4. (2) Let p ≡ ±1 (mod 8). Then T contains maximal subgroups that are isomorphic to S4 and form two conjugacy classes. Let L and R be two subgroups of T which are isomorphic to S4 and are not conjugate. It is easily shown that all subgroups of T of order 3 are conjugate. Hence we may choose L and R such that L ∩ R = S3 . Then Γ = Cos(T , L, R) is T -vertex-biprimitive of valency 4. The subgroups L and R are conjugate in Aut(T ) = PGL2 (p). It follows that Γ is vertex-transitive, and Aut Γ ≥ PGL2 (p). (3) Let G = PGL2 (q), where p ≡ ±11, ±19 (mod 40). Then G contains maximal subgroups that are isomorphic to S4 and all of them are conjugate. Let H be one of them, and let P < H be isomorphic to S3 . Then NG (P ) = S3 × Z2 . Thus there exists an involution g ∈ NG (P ) \ H such that ⟨H , g ⟩ = G and |H : H ∩ H g | = 4. So Cos(G, H , H g ) is G-vertex-biprimitive of valency 4. There are vertex-biprimitive tetravalent graphs associated with groups PGU3 (5) and PGL3 (7). Example 2.10. Let G = PGU3 (5).2 or PGL3 (7).2. By the Atlas [1], G contains two maximal subgroups L ∼ = 32 : 2S4 and R ∼ = 62 : D12 . Both L and R contain Sylow 3-subgroups of G. Thus by Sylow’s Theorem, there exists x ∈ G such that L ∩ Rx contains a Sylow 3-subgroup Q of G. It follows that L ∩ Rx = NL (Q ) = 32 : (2 × S3 ). Hence |L : L ∩ Rx | = |Rx : L ∩ Rx | = 4, and the graph Γ = Cos(G, L, Rx ) is G-vertex-biprimitive of valency 4. Let K = PGU3 (5) < PGU3 (5).2, or K = PGL3 (7) < PGL3 (7).2, respectively. Then K ∩ L = 32 : 2A4 and K ∩ Rx = 62 : S3 . Since both K ∩ L and K ∩ Rx are maximal subgroups of K , the graph Γ is K -vertex-biprimitive. We remark that for G = PGU3 (5).2, the graph Γ is soc(G)-edge-transitive but not soc(G)-vertex-biprimitive, and for G = PGL3 (7).2, Γ is also soc(G)-vertex-biprimitive. The graph displayed in the next example was first given in [2, p. 98]. Example 2.11. The Thompson group G = Th has two maximal subgroups L = [39 ].2S4 and R = 32 .[37 ].2S4 . Both L and R contain Sylow 3-subgroups of G. Then by Sylow’s Theorem there exists x ∈ G such that L ∩ Rx contains a Sylow 3-subgroup Q of G. Thus L ∩ Rx = NL (Q ) = [39 ] : (2 × 3). Then |L : L ∩ Rx | = |Rx : L ∩ Rx | = 4, and the graph Γ = Cos(G, L, Rx ) is G-vertex-biprimitive of valency 4. Finally, we construct examples of vertex-biprimitive graphs of affine type. Example 2.12. Let H be the dihedral group D8 . Then H has a unique faithful irreducible representation of degree 2 over the field Fp , with p > 2 prime. Let G = Z2p : H, and write H = ⟨a⟩ : ⟨b⟩. Obviously, b ∈ GL2 (p) is not a scalar, and it follows that b centralizes an element x of G of order p. Then H ∩ H x = ⟨b⟩, and so the graph Cos(G, H , H x ) is G-vertex-biprimitive, and G-edge-transitive of valency 4. Example 2.13. Let H = A4 . Then H has a unique 3-dimensional faithful irreducible representation over Fp for p > 2. Let G = Z3p : H, and write H = Z22 : ⟨b⟩, where o(b) = 3. It is clear that b ∈ GL3 (p) is not a scalar, and it follows that b centralizes an element x of G of order p. Thus H ∩ H x = ⟨b⟩, and so the graph Cos(G, H , H x ) is G-vertex-biprimitive, G-edge-transitive of valency 4. 3. Local structure of edge-transitive tetravalent graphs In this section, we collect some facts about bipartite edge-transitive tetravalent graphs. Let Γ = (V , E ) be a connected graph. For a vertex α , denote by Γ (α) the neighborhood of α , that is, the set of vertices adjacent to α . For a group G ≤ Aut Γ , the permutation group induced by Gα on Γ (α) is denoted by GΓα (α) , and the kernel of

this action is denoted by G[α1] . Then GΓα (α) ∼ = Gα /G[α1] . For vertices α, β, . . . , γ , let Gαβ...γ = G[α1] ∩ Gβ ∩ · · · ∩ G[γ1] . Assume that Γ is a finite bipartite G-edge-transitive graph of valency 4. Let U and W be the two parts of Γ , and let α ∈ U and β ∈ W be such that {α, β} is an edge. Since Γ is regular of valency 4, we have |Gα | = |Gβ | and |Gα : Gαβ | = |Gβ : Gαβ | = 4. It is clear that both Gα and Gβ are {2, 3}-groups (see also [4]). Further, we have the following [1]

[1]

Lemma 3.1. Assume that Γ is a finite bipartite G-edge-transitive graph of valency 4. Then for each edge {α, β}, one of the following holds: (a) both Gα and Gβ are 2-groups; Γ (β) (b) both GΓα (α) and Gβ are isomorphic to A4 or S4 ;

Γ (β) (c) relabeling if necessary, GΓα (α) ∼ = A4 or S4 , and Gβ ∼ = Z4 , Z22 , or D8 .

C.H. Li, H. Zhang / Discrete Mathematics 317 (2014) 33–43

37

Proof. Assume first that Gα is a 2-group. Since |Gα | = |Gβ |, Gβ is also a 2-group, thus (a) holds. Assume next that 3 | |Gα |. Γ (β)

Notice that each of GΓα (α) and Gβ Γ (β)

is isomorphic to a transitive subgroup of S4 , so we have three possibilities: (1) both GΓα (α) Γ (β)

are isomorphic to A4 or S4 , then in this case (b) holds; (2) both GΓα (α) and Gβ are isomorphic to one of Z4 , Z22 , Γ (β) and D8 . Then it follows that both Gα and Gβ are 2-groups, a contradiction; (3) one of GΓα (α) and Gβ is isomorphic to A4 or and Gβ

S4 , and the other is isomorphic to Z4 , Z22 , or D8 , that is, (c) holds.

Γ (β)

The next example shows that it may happen that the induced groups GΓα (α) and Gβ

are different.

Example 3.2. Let X ∼ = S4 act on U = {α1 , α2 , α3 , α4 }, and let Y ∼ = A4 act on W = {β1 , β2 , β3 , β4 }. Then both actions are transitive. Let G = X × Y , acting naturally on U ∪ W . Then G acts biprimitively on the complete bipartite graph Γ with two parts U and W . Further, Gα1 = Xα1 × Y ∼ = S3 × A4 , and Gβ1 = X × Yβ1 ∼ = S3 × Z3 . So = S4 × Z3 . Hence Gα1 β1 = Xα1 × Yβ1 ∼ Γ (α1 )

G[α11] = S3 , Gβ1 = Z3 , and Gα1 [1]

Γ (β1 )

= A4 , Gβ1

= S4 . Γ (β) Another example is Example 2.6, where Gα = D24 , Gβ = S4 , and GΓα (α) = D8 , Gβ = S4 .

Since the graph Γ is finite and connected, there exists a path of finite length: α0 , α1 , . . . , αn such that G[α10]α1 ...αn = 1. Let α = α0 , and β ∈ Γ (α) \ {α1 }. Then 1] 1] 1] 1 = G[αα ▹ G[αα ▹ . . . ▹ G[αα ▹ G[α1] ▹ Gβα , 1 ...αn 1 ...αn−1 1

Γ (α) Gβα /Gα[1] ∼ = Gβα , and for each i, the factor group 1] 1] 1] 1] ∼ G[αα /G[αα G[1] )/G[α1i+] 1 ∼ )Γ (αi+1 ) ▹ (Gαi αi+1 )Γ (αi+1 ) . = (G[αα = (G[αα 1 ...αi 1 ...αi+1 1 ...αi αi+1 1 ...αi

Γ (β)

Γ (α)

∼ Suppose that GΓα (α) ∼ = Gβ = A4 . Then Gαβ 3-group, leading to the next lemma.

1] ∼ is a 3-group for each i. It follows that Gβα is a = Z3 , and thus G[αα 1 ...αi

Γ (β) Lemma 3.3. Assume that GΓα (α) ∼ = Gβ ∼ = A4 , where {α, β} is an edge. Then Gαβ is a 3-group, and a Sylow 2-subgroup of Gα

is isomorphic to Z22 .

For an integer k ≥ 1, let G[αk] be the pointwise stabilizer of all vertices with distance at most k from α . The largest normal p-subgroup of a group X is denoted by Op (X ). Recall that a graph Γ is called G-locally primitive for some G ≤ Aut Γ if GΓα (α) is primitive for each vertex α of Γ . The following fundamental result, due to J. Van Bon, is an extension of the well-known Thompson–Wielandt theorem on the structure of the point stabilizers of locally primitive graphs. Theorem 3.4 ([13, Theorem 1.1]). Let Γ = (V , E ) be a finite connected G-locally primitive graph. Then for an edge {α, β}, there [1] [1] [2] [3] exists a prime p such that either Gαβ is a p-group, or Gαβ = G[α2] and Gβ = Gβ is a p-group. For tetravalent graphs, we have the following properties. Lemma 3.5. Let Γ = (V , E ) be a connected G-edge-transitive tetravalent graph. Assume that, for an edge {α, β} ∈ E, both Γ (β) ∼ GΓα (α) , Gβ = A4 or S4 . Then either Gα ∼ = A4 or S4 , or one of the following holds: (i) G[α1] ̸= 1, Gαβ = 1, Z3 × A4 ≤ Gα ≤ S3 × S4 , and Gα is 3-local. [1] (ii) 1 ̸= Gαβ is a 2-group, and 33 - |Gα |. [1]

[1]

(iii) 1 ̸= Gαβ is a 3-group, and 25 - |Gα |. (iv) (v)

[2] Gβ [2] Gβ

is a 2-group, and 36 - |Gα |. is a 3-group, and 28 - |Gα |.

Proof. If G[α1] = 1, then Gα acts faithfully on Γ (α) and Gα ∼ = A4 or S4 . [1] Assume next that G[α1] ̸= 1. Assume further that Gαβ = 1. Then Gα = G[α1] .GΓα (α) = (Gαβ .(G[α1] )Γ (β) ).GΓα (α) = (G[α1] )Γ (β) .GΓα (α) . [1]

Γ (β)

Since 1 ̸= (G[α1] )Γ (β) E Gαβ = Z3 or S3 , we have (G[α1] )Γ (β) ∼ = Z3 or S3 . Note that Gαβ is an index 4 subgroup of Gα , we Γ (β) ∼ [1] Γ (α) ∼ conclude that the extension of Gα by Gα splits. Thus Gα = G[α1] × GΓα (α) . Since GΓα (α) , Gβ = A4 or S4 , we know that part (i) holds, and in particular, Gα is 3-local. [1] [1] [2] Suppose now that Gαβ ̸= 1. Then by Theorem 3.4, either Gαβ is a p-group, or Gβ is a p-group, where p = 2 or 3.

Assume that Gαβ is a p-group. Notice that G[α1] = Gαβ .(G[α1] )Γ (β) and (G[α1] )Γ (β) ≤ S3 , if p = 2, then 33 - |Gα |, as in part (ii); if p = 3, then 25 - |Gα |, as in part (iii). Γ (β) Γ (α) Γ (β) Γ (α) [1] [2] [2] Assume that Gαβ is not a p-group. Then Gβ is a p-group. In this case note that Gβ /Gβ ∼ = Gβ2 ≤ Gαβ ≀Gβ , Gαβ ≤ S3 , [1]

Γ (β)

and Gβ

[1]

≤ S4 ; If p = 2, then 36 - |Gα |, as in part (iv). If p = 3, then 28 - |Gβ | = |Gα |, as in part (v).

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4. A reduction Let Γ = (V , E ) be a connected bipartite graph of valency 4 with two parts U and W . Assume that G is transitive on E and intransitive on V , and assume further that G is primitive on both U and W . Suppose that G acts unfaithfully on U. Let K be the kernel of G acting on U. Since G is faithful on V , the kernel K of G acts faithfully on W . Since K E G and G is primitive on W , we conclude that K is transitive on W . It follows that Γ is a complete bipartite graph, and |U | = |W | = 4. Thus K ≥ Z4 or Z22 ≤ K ≤ S4 . Since G is primitive on W , we have the following conclusion. Lemma 4.1. If G is unfaithful on U, then Γ = K4,4 , and Z22 : Z3 ≤ G < Aut Γ = S4 ≀ S2 . In what follows we assume that G is faithful on U. Let N = soc(G) = T1 × · · · × Tk ∼ = T k , with k ≥ 1, and Ti ∼ = T simple. Then N is transitive on both U and W . Lemma 4.2. The socle N is either abelian or non-abelian simple. Proof. Suppose that k > 1 and T is non-abelian. For α ∈ U, since 1 ̸= Nα E Gα , it is readily seen that Γ is an N-edgetransitive tetravalent graph. Let M = T1 × · · · × Tk−1 . Then M ▹ N. Since Nα is a {2, 3}-group, M is intransitive on U. Then the quotient graph ΓM is N /M-edge-transitive and has valency dividing 4. Since T = N /M ≤ Aut ΓM , the valency of ΓM is 4, and hence M is semiregular on U. By the O’Nan–Scott Theorem (see [3]), G is not primitive on U, which is a contradiction. Thus either k = 1 and N is non-abelian simple, or N is abelian. By Lemma 4.2, when G is primitive on both U and W , then the primitive type is affine or almost simple. First we deal with the affine case, that is, N is abelian. Let N = soc(G) = Znp , with p prime and n ≥ 1. Then N is a regular normal subgroup of G, and we may identify both U and W with N. Let α be the zero vector. Then G = N : Gα , and Gα is an irreducible subgroup of GLn (p). By [12, Lemma 9], Gα is faithful on Γ (α), and so Gα = Z4 , Z22 , D8 , A4 or S4 . For these cases, all the faithful irreducible representations of Gα are known, see [11] for example. In particular, the group Z22 has no faithful irreducible representation (this also follows from Schur’s lemma). This gives the following lemma. Lemma 4.3. If soc(G) = Znp , then the group G is one of the following groups:

Zp : Z4

with p ≡ 1 (mod 4),

Z2p : Z4

with p ≡ 3 (mod 4),

Z2p : D8 , Z3p : A4

and

Z3p : S4 .

Proof. As mentioned before, the possibilities for Gα are Z4 , D8 , A4 and S4 . Assume that Gα = Z4 , and G = Zp : Z4 or Z2p : Z4 . Then none of the non-identity elements of Op (G) is centralized by a non-identity element of Gα . Thus for an element x ∈ Op (Gα ) of order p, the intersection Gα ∩ Gxα = 1, so the graph Cos(G, Gα , Gxα ) is G-vertex-biprimitive and G-edge-transitive of valency 4. For G = Z2p : D8 or G = Z3p : A4 , Examples 2.12 and 2.13 show the existence of G-vertex-biprimitive and G-edge-transitive tetravalent graphs. Finally, assume that Gα = S4 , and G = Z3p : S4 . Let K < Gα be isomorphic to S3 . It is easily shown that K centralizes a non-identity element x ∈ Op (G). Then Gα ∩ Gxα = K , and the coset graph Cos(G, Gα , Gxα ) is G-vertex-biprimitive, G-edgetransitive, and of valency 4. Remark. The graphs corresponding to the groups of Lemma 4.3 were constructed (by a different approach) as bipartite graphs or the standard double covers of certain Cayley graphs in [5]. 5. Almost simple type We treat the almost simple case in this section. Let Γ = (V , E ) be a connected graph of valency 4. Let {α, β} be an edge of Γ . We first prove a technical lemma which will be used frequently. Lemma 5.1. Assume that Gα , Gβ are conjugate (in G), and the subgroups of Gα which are isomorphic to Gαβ are all conjugate (in Gα ). Then there exists g ∈ NG (Gαβ ) such that Gβ = Ggα . In particular, NG (Gαβ ) ̸⊂ Gα ∪ Gβ . Proof. Since Gα , Gβ are conjugate, there exists x ∈ G such that Gxα = Gβ . Then Gαβ ∼ = Gxαβ < Gxα = Gβ . Thus there exists h x h g xh h ∈ Gβ such that Gxh αβ = (Gαβ ) = Gαβ . Now g := xh ∈ NG (Gαβ ) is such that Gα = Gα = Gβ = Gβ .

Assume that Γ is bipartite with two parts U and W , such that G ≤ Aut Γ is transitive on E and intransitive on V . As noticed before, Gα and Gβ are {2, 3}-groups. Assume further that G is almost simple and primitive on both U and W . Then Gα and Gβ are maximal subgroups of G. Moreover, since Γ is of valency 4, the intersection Gαβ = Gα ∩ Gβ has index 4 in both Gα and Gβ .

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The pairs (G, H ), with G almost simple and H = Gα or Gβ , are known, see for instance [10], with a long list of candidates. While in the case that Gα and Gβ are conjugate, the insoluble primitive permutation groups which have a suborbit of length 4 are shortly listed in [9, Theorem 3.4]. Our task is to identify all the triples (G, Gα , Gβ ) from these sources. We first deal with the case where Gα is a 2-group. Lemma 5.2. Let Gα be a 2-group. Then G = PGL2 (7), PGL2 (9), M10 , PΓ L(2, 9) or PSL(2, 17), and Γ is the standard double cover of an arc-transitive graph. Proof. Since Gα , Gβ are both maximal 2-subgroups of G, it follows that Gα , Gβ are Sylow 2-subgroups of G, and hence they are conjugate. Therefore by [4, Lemma 2.5], Γ is the standard double cover of a vertex-primitive arc-transitive graph of valency 4 (directed or undirected). Since subgroups of G which are isomorphic to Gα are conjugate, the pairs (G, Gα ) must be lying in the list of insoluble primitive permutation groups with a suborbit of length 4 given in [9, Theorem 3.4]. Inspecting the list, we conclude that the pair (G, Gα ) is one of the following: G Gα

PGL2 (7) D16

PGL2 (9) D16

M10 Z8 :Z2

PΓ L(2, 9) [25 ]

PSL(2, 17) D16

By [9, Theorem 1.5], each pair (G, Gα ) corresponds to a vertex-primitive arc-transitive graph of valency 4, hence the result follows. We thus assume that 3 divides |Gα | = |Gβ |. Lemma 5.3. If Gα ∼ ̸ Gβ , then (G, Gα , Gβ ) lies in the following table: = G Gα Gβ

PGL2 (11) S4 D24

PGL2 (13) S4 D24

PSL2 (23) S4 D24

PGL3 (7) 32 :2A4 62 :S3

PSp4 (3) [33 ]:2A4 [33 ]:S4

PGU3 (5) 32 :2A4 62 :S3

Th [39 ].2S4 32 .[37 ].2S4

Moreover, there is a unique graph corresponding to each case. Proof. Since Gα ∼ ̸ Gβ , the case (b) or (c) of Lemma 3.1 holds. Notice that |Gα | = |Gβ |. Inspecting [10], we found that the = candidates of the triples (G, Gα , Gβ ) are as follows: (PGL2 (11), D24 , S4 ), (PSL2 (13), D12 , A4 ), (PGL2 (13), D24 , S4 ), (PSL2 (23), D24 , S4 ), (PGL3 (7), [32 ] : 2A4 , [32 ] : S4 ), (PGL3 (7).2, 32 : 2S4 , 62 : D12 ), (PSp4 (3), [33 ] : 2A4 , [33 ] : S4 ), (PSp4 (3).2, [33 ] : 2S4 , [33 ] : (S4 × 2)), (PGU3 (5), 32 : 2A4 , 62 : S3 ), (PGU3 (5).2, 32 : 2S4 , 62 : D12 ), and (Th, [39 ].2S4 , 32 .[37 ].2S4 ). By Examples 2.6, 2.7, 2.10 and 2.11, for each of the above cases, there exists a G-vertex-biprimitive, G-edge-transitive tetravalent graph Γ . Further, as indicated in Examples 2.6, 2.7 and 2.10, those groups with the same socle correspond to the same graph. By the Atlas [1], all subgroups of G that are isomorphic to Gα are conjugate, and hence we can fix Gα . Further, in all these cases, let K be a subgroup of Gα of index 4 that could be the possible arc stabilizer Gαβ . Then K = S3 or Z3 , and so subgroups of Gα which are isomorphic to K form a unique class in Gα . Thus we may fix such a subgroup K . Then the candidates for Gβ are subgroups R < G such that R ̸= Gα and R > K . Let R1 , R2 be two such subgroups. Then R1 , R2 > K and Rx1 = R2 for some h x ∈ G. Now K , K x < R2 , and so K = K xh for some h ∈ R2 . Thus xh ∈ NG (K ), and Rxh 1 = R2 = R2 . Notice that K = Q : 2 or Q : [4], where Q is a Sylow 3-subgroup of G. It follows from the information given in the Atlas [1] that NG (K ) ≤ NG (Q ) < R1 . So R1 = Rxh 1 = R2 , and hence there is only one corresponding graph. Lemma 5.4. If Gα ∼ = Gβ and Gα , Gβ are not conjugate in G, then Γ is vertex-transitive, and (G, Gα , Gβ ) lies in the following table: G ∼ Gβ Gα =

M12 32 :2S4

PSL3 (3) 32 :2S4

G2 (3) (31++2 × 32 ):2S4

PSL2 (p) with p ≡ ±1 (mod 8) S4

Moreover, in each case the corresponding graph is unique. Proof. Assume that the maximal subgroups of G that are isomorphic to Gα form two conjugacy classes C1 and C2 . Then by [10, Table A], (G, Gα ) is one of the following pairs: (A6 , S4 ), (M12 , 32 : 2S4 ), (PSL3 (3), 32 : 2S4 ), (G2 (3), (31++2 × 32 ) : 2S4 ), or (PSL2 (p), S4 ), where p ≡ ±1 (mod 8). For (G, Gα , Gβ ) = (A6 , S4 , S4 ) with Gα and Gβ being not conjugate, the intersection Gα ∩ Gβ should be isomorphic to S3 , which is not possible. For each of the other cases, the existence of Γ is shown in Example 2.5 or 2.9. Suppose that Gα ∈ C1 . Since all subgroups of Gα of index 4 are conjugate in Gα , we may choose such a subgroup K (as potential Gαβ ). Then candidates for Gβ are subgroups R ∈ C2 which contains K . Arguing as in the proof of Lemma 5.3, we conclude that there is only one suitable graph. We still need to treat the case where Gα and Gβ are conjugate in G. As indicated before, in this case the pairs (G, Gα ) are also in the list of insoluble primitive permutation groups which have a suborbit of length 4, given in [9, Theorem 3.4]. In what follows instead of using [10], we will use this list. First we prove the following result.

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Lemma 5.5. Assume that all maximal subgroups of G that are isomorphic to Gα are conjugate (in G), and all subgroups K of Gα of index 4 are conjugate and maximal (in Gα ). Then there exist exactly |NG (K )/K | − 1 different graphs which are G-edge-transitive and G-vertex-biprimitive of valency 4. In particular, if NG (K )/K ∼ = Z2 , then there is a unique graph Cos(G, Gα , Gβ ), which is the standard double cover of Cos(G, Gα , Gα gGα ), where g ∈ NG (K ) \ K . Proof. Since maximal subgroups of G that are isomorphic to Gα are all conjugate, we can fix Gα . Let K be a fixed subgroup of Gα of index 4. By Lemma 5.1, any suitable graph has the form Cos(G, Gα , Ggα ), for some element g ∈ NG (K ) \ K . On the other hand, each element g ∈ NG (K ) \ K gives rise to a graph Γ = Cos(G, Gα , Ggα ), which is G-edge-transitive and G-vertex-biprimitive graph of valency 4. g g g g For two elements g1 , g2 ∈ NG (K ) such that g2 ∈ Kg1 , we have Gα1 = Gα2 , and so Cos(G, Gα , Gα1 ) = Cos(G, Gα , Gα2 ). g

g1 g

g

−1

Conversely, assume that Gα1 = Gα2 , where g1 , g2 ∈ NG (K ). Then Gα 2 = Gα . Since Gα is maximal in G, we have g1 g2−1 ∈ Gα , and hence g1 g2−1 ∈ Gα ∩ NG (K ) = NGα (K ). Moreover, since K is maximal in Gα , we conclude that NGα (K ) = K , and so g1 ∈ Kg2 . Therefore, there exist exactly |NG (K )/K | − 1 different graphs. Assume that Gα and Gβ are conjugate in G. Then by [9, Theorem 3.4] (note that the assumption that 3 divides |Gα |), the candidates are as follows: (1) (2) (3) (4) (5)

Gα ∼ = A4 , and G = A5 or PSL2 (q), where q = p ≡ ±13, ±37 (mod 40) or q = 3f with f ≥ 3 prime. Gα ∼ = S4 , and G = PSL3 (3) or PSL2 (p) with p ≡ ±1 (mod 8), or PGL2 (p) with p ≡ ±11, ±19 (mod 40). (G, Gα ) = (PΣ L2 (27), A4 × Z3 ). (G, Gα ) = (PSL3 (7), (A4 × Z3 ).Z2 ). (G, Gα ) = (A7 , (A4 × Z3 ).Z2 ) or (S7 , S4 × S3 ). The first two cases correspond to graphs with G[α1] = 1, that is, Gα acts faithfully on Γ (α).

Lemma 5.6. Assume that Gα , Gβ are conjugate and isomorphic to S4 . Then G = PSL2 (p), with p ≡ ±1 (mod 8), or G = PGL2 (p) with p ≡ ±11, ±19 (mod 40), and Γ is the standard double cover of an arc-transitive graph. Proof. By the above discussion, in this case we have (G, Gα ) = (PSL3 (3), S4 ), (PSL2 (p), S4 ) with p ≡ ±1 (mod 8), or (PGL2 (p), S4 ), where p ≡ ±11, ±19 (mod 40). For the case (G, Gα ) = (PSL3 (3), S4 ), by Example 2.8, there is a G-biprimitive tetravalent graph, which is not the standard double cover of a vertex-primitive tetravalent graph. This graph has been listed in Table A. For the remaining two cases, notice that Gα and Gβ are conjugate, Gαβ ∼ = S3 , and all the subgroups of Gα that are isomorphic to Gαβ are conjugate. By Lemma 5.1, there exists g ∈ NG (Gαβ ) \ Gα such that Gβ = Ggα . Furthermore, NG (Gαβ ) = Gαβ × Z2 , and NG (Gαβ )/Gαβ = Z2 . By Lemma 5.5, there is a unique graph corresponding to this case, which is the standard double cover of a vertex-primitive, arc-transitive tetravalent graph. Lemma 5.7. Assume that Gα , Gβ are conjugate and isomorphic to A4 . Then either G = A5 , Γ = K5,5 − 5K2 , or G = PSL(2, q), and Γ is the standard double cover of an undirected graph or a digraph, where either (i) q = p ≡ ±13, ±37 (mod 40), and there are exactly f −1

f

(ii) q = 3 with f ≥ 3 prime, and there are exactly 3 of valency 4.

p+ϵ 3

− 1 different suitable graphs; or − 1 different graphs that are G-vertex-biprimitive and G-edge-transitive

Proof. In this case the candidates for (G, Gα ) are (A5 , A4 ), and (PSL2 (q), A4 ), where q = p ≡ ±13, ±37 (mod 40), or q = 3f with f ≥ 3 prime. If (G, Gα ) = (A5 , A4 ), then Γ = K5,5 − 5K2 , which is the standard double cover of the graph K5 . Next we consider the case where (G, Gα ) = (PSL2 (q), A4 ). Notice that Gα and Gβ are conjugate, Gαβ ∼ = Z3 , and all subgroups of Gα that are isomorphic to Gαβ are conjugate and maximal in Gα . By Lemma 5.1, there exists g ∈ NG (Gαβ ) \ Gα such that Gβ = Ggα . Let q = p ≡ ±13, ±37 (mod 40). Then NG (Gαβ ) ∼ = Dq+ε , where 3 divides q + ε . For each element g ∈ NG (Gαβ ) \ Gαβ , the subgroup Ggα is such that Cos(G, Gα , Ggα ) is G-vertex-biprimitive and G-edge-transitive of valency 4. Therefore by Lemma 5.5, q+ε there are exactly 3 − 1 different graphs. For the case where q = 3f with f

≥ 3 prime, the normalizer NG (Gαβ ) = Zf3 is a Sylow 3-subgroup of G, so − 1 different graphs of valency 4 which are G-vertex-biprimitive

f −1 Z3 . By Lemma 5.5, there are exactly 3f −1

NG (Gαβ )/Gαβ ∼ = and G-edge-transitive.

Finally we deal with the remaining three cases. For each case, if there exists a corresponding graph Γ , then Gα must be unfaithfully on Γ (α), that is, G[α1] ̸= 1. Lemma 5.8. Assume that Gα and Gβ are conjugate in G and G[α1] ̸= 1. Then one of the following holds: (i) (G, Gα ) = (A7 , (A4 × Z3 ).Z2 ) or (S7 , S4 × S3 ), and Γ is the standard double cover of the odd graph O3 . (ii) (G, Gα ) = (PSL3 (7), (A4 × Z3 ).Z2 ), and there is only one corresponding graph.

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Proof. Suppose that (G, Gα ) = (PΣ L2 (27), A4 × Z3 ). Then Gαβ = Z23 is a Sylow-3-subgroup of Gα , and so subgroups of Gα which are isomorphic to Gαβ are conjugate. If there exists a G-vertex-biprimitive tetravalent graph Γ = Cos(G, Gα , Gβ ), where Gβ = Ggα for some g ∈ G, then by Lemma 2.4, it is the standard double cover of the graph Σ = Cos(G, Gα , Gα gGα ), and by Lemma 5.1, the element g can be chosen such that g ∈ NG (Gαβ ). By the Atlas [1], the subgroup NG (Gαβ ) is contained in a maximal subgroup of G which is isomorphic to Z33 : Z13 : Z3 , which means that Gα gGα ̸= Gα g −1 Gα . It follows that the valency of Σ is 8, a contradiction. Therefore this case gives no biprimitive tetravalent graph. Suppose that (G, Gα ) = (A7 , (A4 × Z3 ) : Z2 ) or (S7 , S4 × S3 ). Then for the former case, Gαβ = Z23 : Z2 , and by the Atlas [1], NG (Gαβ ) = Z23 : Z22 , and NG (Gαβ )/Gαβ = Z2 . By Lemma 5.5, Γ is the standard double cover of a vertex-primitive, arctransitive tetravalent graph, which is actually the odd graph O3 . It is then easily shown that the pair (G, Gα ) = (S7 , S4 × S3 ) gives rise to the same graph. For (G, Gα ) = (PSL3 (7), (A4 × Z3 ) : Z2 ), we have Gαβ ∼ = S3 × Z3 , and by the Atlas [1], we have NG (Gαβ ) = 32 : Q8 . Thus, NG (Gαβ )/Gαβ ∼ = Z2 , and by Lemma 5.5, Cos(G, Gα , Ggα ) is the unique G-vertex-biprimitive tetravalent graph, where g ∈ NG (Gαβ ) \ Gα . 6. Proof of Theorem 1.1 In this final section, we will complete the proof of the main theorem. All G-vertex-biprimitive edge-transitive graphs of valency 4 are listed in Theorem 1.1, shown in the last two sections. Our task here is to determine their automorphism groups and isomorphism classes. We first prove a simple lemma. Lemma 6.1. Let Γ = (V , E ) be a connected bipartite graph with two parts U and W . Then V = U ∪ W is the only bipartition of V . Proof. Suppose that V = U ′ ∪ W ′ is another bipartition. Then V = (U ∩ U ′ ) ∪ (U ∩ W ′ ) ∪ (W ∩ U ′ ) ∪ (W ∩ W ′ ) is a partition of V . It follows that no vertex in (U ∩ W ′ ) ∪ (W ∩ U ′ ) is adjacent to a vertex in (U ∩ U ′ ) ∪ (U ′ ∩ W ′ ), which is a contradiction since Γ is connected. Let Γ = (V , E ) be a connected bipartite graph of valency 4 with parts U and W . Assume that G is transitive on E and intransitive on V , and assume further that G is primitive on U and W . Then G = GU = GW , and G ∼ = GU ∼ = GW . Let A = Aut Γ . + By Lemma 6.1, A preserves the bipartition V = U ∪ W . Let A = AU = AW . Then for an edge {α, β}, we have G ≤ A+ , Gα ≤ A+ α + + + + and Gβ ≤ A+ β . Therefore, if A is faithful on U (and so on W ), then (A , Aα , Aβ ) is one of the triples (G, Gα , Gβ ) satisfying Lemmas 4.3, 5.2–5.4, 5.6–5.7, and 5.8. An inspection of these triples leads to the following lemma. Lemma 6.2. If A+ is faithful on U, then either (1) G = Z5 : Z4 , A+ = S5 , and Γ = K5,5 − 5K2 , or (2) G is normal in A. Proof. Suppose that G is soluble and A+ is insoluble. It follows from Lemmas 5.2–5.4, and 5.6–5.8 that A+ = S5 , G = Z5 : Z4 , and Γ = K5,5 − 5K2 , as in part (1). For the other cases, by Lemmas 4.3, 5.2–5.4, and 5.6–5.8, we conclude that G is normal in A. We next determine A = Aut Γ . Lemma 6.3. Assume that soc(G) is abelian. Then Γ is arc-transitive, and either part (i) or part (ii) of Theorem 1.1 holds. Proof. Let R = soc(G). Then G = R : Gα . By Lemma 4.3, we have R = Zp , with Gα = Z4 ; or R = Z2p , with Gα = Z4 or D8 ; or R = Z3p , with Gα = A4 or S4 . Let A = Aut Γ , and let A+ = AU = AW . If A+ is unfaithful on U, then by Lemma 4.1, we conclude that A = S4 ≀ S2 , as in part (i) of Theorem 1.1. Thus, we assume that A+ is faithful on U. If A+ is insoluble, by Lemma 6.2, part (i) of Theorem 1.1 holds. We hence further assume that A+ is soluble. By Lemma 4.3, if G = Zp : Z4 with p > 5, or G = Z2p : D8 , or Z3p : S4 , then A+ = G. Next consider the cases where G = Z2p : Z4 and G = Z3p : A4 . First, let G = Z2p : Z4 < AGL(2, p) with p ≡ 3 (mod 4). Then there exists g ∈ G of order p such that Γ = Cos(G, Gα , Ggα ).

The group G has an over group X = Z2p : D8 in AGL(2, p). Write Xα = Gα : ⟨b⟩. Then b centralizes an element x ∈ Z2p \ {1}. Let ρ ∈ GL(2, p) be such that bρ = b and g = xρ . Then bg = gb. Let Y = X ρ . Then Yα ∩ Yαg = ⟨b⟩, and hence Cos(G, Gα , Ggα ) = Cos(Y , Yα , Yαg ). So A+ = Z2p : D8 . Next let G = Z3p : A4 , and Gα = Z22 : ⟨z ⟩. Then z centralizes a unique cyclic subgroup ⟨g ⟩ of order p, and

Γ = Cos(G, Gα , Ggα ). In AGL(3, p), there is an overgroup X = Z3p : S4 of G. Then Xα = Gα : ⟨b⟩ = Z22 : ⟨z , b⟩, and ⟨g ⟩ is the unique cyclic subgroup of G of order p which is centralized by ⟨z , b⟩. Thus, Xα ∩ Xαg = ⟨z , b⟩, and hence Cos(G, Gα , Ggα ) = Cos(X , Xα , Xαg ). So A+ = Z3p : S4 .

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Finally, by [8], Γ is arc-transitive, and hence A = A+ .Z2 . It follows that one of the following cases occurs: G = Zp : Z4 , and A = A+ .Z2 = (Zp : Z4 ).Z2 ∼ = (Zp : Z4 ) × Z2 . G = Z2p : Z4 or Z2p : D8 , and A = A+ .Z2 = (Z2p : D8 ).Z2 ∼ = (Z2p : D8 ) × Z2 . 3 3 + 3 ∼ G = Zp : A4 or Zp : S4 , and A = A .Z2 = (Zp : S4 ).Z2 = (Z3p : S4 ) × Z2 . Each of them satisfies part (ii) of Theorem 1.1.

Next we determine the isomorphism classes of Γ for almost simple case. Assume that G is almost simple. By Lemmas 5.2–5.4, and 5.6–5.8, either Γ is unique, or G = PSL(2, q) and Gα ∼ = Gβ ∼ = A4 . Lemma 6.4. Let G = PSL(2, q), and Gα , Gβ be conjugate and isomorphic to A4 , as in Lemma 5.7. Then the following hold: (i) For q = p ≡ ±13, ±37 (mod 40), there are exactly 6 non-isomorphic graphs that are G-vertex-biprimitive and G-edgetransitive, where ε = 1 or −1 such that 3 | (p + ε). (ii) For q = 3f with f ≥ 3 prime, there are exactly (3f −1 − 1)/2 non-isomorphic suitable graphs of valency 4. p+ε

Proof. Since all subgroups of G that are isomorphic to A4 are conjugate, we may fix Gα . Let K be a Sylow 3-subgroup of Gα . By Lemma 5.7, there are exactly n different graphs:

Γi = Cos(G, Gα , Ggαi ), with gi ∈ NG (K ) such that gi K ̸= gj K if 1 ≤ i < j < n. Here n = 3 − 1 or 3f −1 − 1, corresponding to q = p ≡ ±13, ±37 (mod 40), or q = 3f , respectively. Assume that Γi ∼ = Γj , where i < j. Let σ be an isomorphism between Γi and Γj . Then Gσ ≤ (Aut Γi )σ = Aut Γj . By Lemma 6.2, Aut Γj is almost simple and contains a normal subgroup G. It follows that Gσ is normal in Aut Γj , and hence Gσ = G. Thus, σ induces an automorphism of G, namely, p+ε

gj

Cos(G, Gσα , (Ggαi )σ ) = Γiσ = Γj = Cos(G, Gα , Gα ). gj

By Lemma 2.2, Gσα = Ggα and (Gαi )σ = (Gα )g , where g ∈ G. Write τ = σ g −1 . Then, τ ∈ NAut(G) (Gα ) = S4 and τ maps Gαi to gj Gα . It follows that τ is an involution and normalizes a Sylow 3-subgroup of Gα . Without loss of generality, we may assume that τ normalizes K . Hence τ ∈ NAut(G) (K ) = NG (K ) : ⟨τ ⟩. First, we consider the case where q = p ≡ ±13, ±37 (mod 40). We observe that the elements of NAut(G) (K ) = D2(p+ε) that are centralized by τ lie in the center Z(NAut(G) (K )) = Z2 . It follows that τ centralizes one involution of NG (K )/K and p+ε−6 p+ε divides the other 3 − 2 non-identity elements of NG (K )/K into two orbits of size 6 . Therefore, in NG (K )/K , there are g

g

+ 1 images that are not conjugate under τ , which equals p+ε . 6 f f −1 For q = 3 with f ≥ 3 prime, the normalizer NG (K ) = Z3 is a Sylow 3-subgroup of G, and NG (K )/K ∼ = Z3 . Notice f −1 f that NAut(G) (K ) = NG (K ) : ⟨τ ⟩ = Z3 : Z2 such that τ inverts all elements of NG (K ). It follows that there are exactly 3 2 −1 exactly

p+ε−6 6

f

non-identity elements of NG (K )/K that are not conjugate under τ . So there are exactly

3f −1 −1 2

non-isomorphic graphs.

Now we are ready to prove Theorem 1.1. Proof of Theorem 1.1. If G is unfaithful on U, then Lemma 4.1 shows that Γ = K4,4 , as in part (i) of Theorem 1.1. Assume that G is faithful on both U and W . By Lemma 4.2, the socle soc(G) is abelian or simple. Moreover, if soc(G) is abelian, then Lemma 6.3 shows that Γ is as in part (i) or (ii) of Theorem 1.1. We may thus assume that soc(G) is nonabelian simple, namely G is almost simple. Let Γ = Cos(G, Gα , Gβ ), where {α, β} is an edge. If both Gα and Gβ are 2-groups, then Lemma 5.2 tells that Γ is the standard double cover of a vertex-primitive arc-transitive graph of valency 4, as in part (iii) of Theorem 1.1. Suppose next that Gα and Gβ are not 2-groups. If in addition Gα and Gβ are not conjugate in G, then by Lemmas 5.3 and 5.4, the triple (G, Gα , Gβ ) and Γ satisfy part (vi) of Theorem 1.1. The remaining case is that Gα and Gβ are conjugate. Then either Gα and Gβ are A4 or S4 , or Gα is a 3-local group. If Gα ∼ = S4 , then Lemma 5.6 shows that G = PSL2 (p) with p ≡ ±1 (mod 8) or G = PGL2 (p), with p ≡ ±11, ±19 (mod 40), and Γ is the standard double cover of a vertex-primitive arc-transitive graph of valency 4, as in part (iii) of Theorem 1.1. If Gα ∼ = A4 , then by Lemmas 5.7 and 6.4, the statements of part (iv) and part (v) of Theorem 1.1 are true. Finally, if Gα is 3-local, then Lemma 5.8 shows that either (G, Gα ) = (A7 , (A4 × Z3 ).Z2 ) or (PSL3 (7), (A4 × Z3 ).Z2 ), and Γ is the standard double cover of a vertex-primitive arc-transitive graph of valency 4, or (G, Gα ) = (PΣ L2 (27), A4 × Z3 ) or (PSL2 (27).6, S4 × Z3 ), and Γ is a G-vertex-biprimitive graph given in Lemma 5.7, (ii), as in part (iii) or (V) of Theorem 1.1. Acknowledgments The authors would like to express their heartfelt thanks to the referees for numerous suggestions that improved the exposition, and they would also express their special thanks to an anonymous referee who pointed out a missing case in the main result.

C.H. Li, H. Zhang / Discrete Mathematics 317 (2014) 33–43

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