¿^creative , ars mathematica ^commons contemporánea Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 10 (2016) 269-280 2-Arc-Transitive regular covers of Kn,n — nK2 with the covering transformation group Zp Wenqin Xu, Yanhong Zhu , Shaofei Du * School of Mathematical Sciences,Capital Normal University, Beijing, 100048, P R China Received 27 January 2014, accepted 24 December 2015, published online 12 January 2016 Abstract In 2014, Xu and Du classified all regular covers of a complete bipartite graph Kn,n minus a matching, denoted by Kn n - nK2, whose covering transformation group is cyclic and whose fibre-preserving automorphism group acts 2-arc-transitively. In this paper, a further classification is achieved for all the regular covers of Kn n - nK2, whose covering transformation group is isomorphic to Zp with p a prime and whose fibre-preserving automorphism group acts 2-arc-transitively. Actually, there are only few covers with these properties and it is shown that all of them are covers of K4,4 - 4K2. Keywords: Arc-transitive graph, covering graph, 2-transitive group. Math. Subj. Class.: 05C25, 20B25, 05E30 1 Introduction Throughout this paper graphs are finite, simple and undirected. For the group- and graph-theoretic terminology we refer the reader to [15, 17]. For a graph X, let V(X), E(X), A(X) and Aut X denote the vertex set, edge set, arc set and the full automorphism group of X respectively. An edge and an arc of X are denoted by {u, v} and (u, v), respectively. An s-arc of X is a sequence (v0, vi,..., vs) of s +1 vertices such that (vj, vj+1) G A(X) and vj = vi+2, and X is said to be 2-arc-transitive if Aut X acts transitively on the set of 2-arcs of X. Let X be a graph, and let P be a partition of V(X) into disjoint sets of equal size m. The quotient graph Y := X/P is the graph with the vertex set P and two vertices P1 and P2 of Y are adjacent if there is at least one edge between a vertex of P1 and a vertex of * corresponding author E-mail addresses: wenqinxu85@163.com (Wenqin Xu), zhuyanhong911@126.com (Yanhong Zhu), dushf@mail.cnu.edu.cn (Shaofei Du) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 270 Ars Math. Contemp. 10 (2016) 255-268 P2 in X. We say that X is an m-fold cover of Y if the edge set between Pi and P2 in X is a matching whenever P\P2 € E(Y). In this case Y is called the base graph of X and the sets Pi are called the fibres of X. An automorphism of X which maps a fibre to a fibre is said to be fibre-preserving. The subgroup K of all those automorphisms of X which fix each of the fibres setwise is called the covering transformation group. It is easy to see that if X is connected then the action of K on the fibres of X is necessarily semiregular, that is, Kv = 1 for each v € V (X). In particular, if this action is regular we say that X is a regular cover of Y. The main motivation for the present paper is to contribute toward the classification of finite 2-arc-transitive graphs. In [23, Theorem 4.1], Professor Praeger divided all the finite 2-arc-transitive graphs X into the following three subclasses: (1) Quasiprimitive type: every nontrivial normal subgroup of Aut X acts transitively on vertices; (2) Bipartite type: every nontrivial normal subgroup of Aut X has at most two orbits on vertices and at least one of them has two orbits on vertices; (3) Covering type: there exists a normal subgroup of Aut X having at least three orbits on vertices, and thus X is a regular cover of some graphs of types (1) or (2). During the past twenty years, a lot of results regarding the primitive, quasiprimitive and bipartite 2-arc-transitive graphs have appeared [11, 18, 19, 20, 23, 24]. However, very few results concerning the 2-arc-transitive covers are known, except for some covers of graphs with small valency and small order. The first meaningful class of graphs to be studied might be complete graphs. In [7], a classification of covers of complete graphs is given, whose fibre-preserving automorphism groups act 2-arc-transitively and whose covering transformation group is either cyclic or Zp. This classification is generalized in [8] to covering transformation group Zp. In [26], the same problem as in [7] and [8] is considered, but the covering transformation group considered is metacyclic. As for covers of bipartite type, in [25], all regular covers of complete bipartite graph minus a matching Kn n - nK2 were classified, whose covering transformation group is cyclic and whose fibre-preserving automorphism group acts 2-arc-transitively. In this paper, we consider the same base graphs while the covering transformation group is Zp with p a prime. Remarkably, we shall show that all the regular covers with these properties are just covers of K4,4 - 4K2. Note that to classify regular covers of given graphs such as Kn and Kn,n, whose covering transformation group is an elementary group Zp and whose fibre-preserving automorphism group acts 2-arc-transitively is a very difficult task. Essentially, it is related to the group extension theory, the group representation theory and other specific branches of group theory. We believe that the classification of all such covers for all the values k is almost not feasible. Therefore, the first step might be to study the problem for small values k and to construct some new interesting covers. Except for the graph Kn n - nK2, another often considered graph is the complete bipartite graph Kn n. In further research, we shall focus on the 2-arc-transitive regular elementary abelian covers of this graph. For further reading on the topic of covers, see [4, 5, 9, 13, 14, 22]. A cover of a given graph can be derived through a voltage assignment, see Gross and Tucker [15, 16]. Let Y be a graph and K a finite group. A voltage assignment (or, K-voltage assignment) on the graph Y is a function f : A(Y) ^ K with the property that W. Xu et al.: 2-Arc-Transitive regular covers of Kn,n — nK2... 271 f (u, v) = f (v, u)- for each (u, v) G A(Y). The values of f are called voltages, and K is called the voltage group. The derived graph Y Xf K from a voltage assignment f has for its vertex set V(Y) x K, and its edge set {{(u, g), (v, f (v, u)g)} I {u, v} G E(Y), g G K}. By the definition, the derived graph Y Xf K is a covering of the graph Y with the first coordinate projection p : Y x f K ^ Y, which is called the natural projection and with the covering transformation group isomorphic to K. Conversely, each connected regular cover X of Y with the covering transformation group K can be described by a derived graph Y x f K from some voltage assignment f. Moreover, the voltage assignment f naturally extends to walks in Y. For any walk W of Y, let fW denote the voltage of W. Finally, we say that an automorphism a of Y lifts to an automorphism a of X if ap = pa, where p is the covering projection from X to Y. Before stating the main result, we first introduce a family of derived graphs. Let Y = K4,4 - 4K2 with the bipartition V(Y) = {a, 6, c, d} U {w, x, y, z} as shown in Figure (a), and fix a spanning tree T of K4 4 - 4K2 as shown in Figure (b). Identify the elementary group Zp with the 2-dimensional linear vector space over Fp. Then we define a family of derived graphs X(p) := (K4,4 - 4K2) x^ Zp with voltage assignment ^ such that ^(6, y) = (1, 0), ^(c, w) = ^(d, w) = ^(d, x) = (0,1), ^(c, x) = (1,1) and ^(w,v) = 0 for any tree arc (u, v). Figure (a): the graph K4,4 — 4K2; (b): a spanning tree T of K4,4 — 4K2. a w a w The following theorem is the main result of this paper. Theorem 1.1. Let X be a connected regular cover of the complete bipartite graph minus a matching Kn,n — nK2 (n > 3), whose covering transformation group K is isomorphic to Zp with p a prime and whose fibre-preserving automorphism group acts 2-arc-transitively. Then n = 4 and X is isomorphic to X(p). 2 Preliminaries In this section we introduce some preliminary results needed in Section 3. The first result may be deduced from the classification of doubly transitive groups (see [2] and [3, Corollary 8.3]). Proposition 2.1. Let G be a 3-transitive permutation group of degree at least 4. Then one of the following occurs. 272 Ars Math. Contemp. 10 (2016) 255-268 (i) G S4; (ii) soc(G) is 4-transitive; (iii) soc(G) = M22 or A5, which are 3-transitive but not 4-transitive; (iv) PSL(2, q) < G < PrL(2, q), where the projective special linear group PSL(2, q) is the socle of G which does not act 3-transitively, and G acts on the projective geometry PG(1, q) in a natural way, having degree q +1, with q > 5 an odd prime power; (v) G = AGL(m, 2) with m > 3; (vi) G = Z4 x A7 < AGL(4, 2). Let G be a finite group and H be a proper subgroup of G, and let D = D-1 be inverse-closed union of some double cosets of H in G \ H. Then the coset graph X = X (G; H,D) is defined by taking V(X) = {Hg | g g G} as the vertex set and E(X) = {{Hg1,Hg2} | g2g-1 G D} as the edge set. By the definition, the size of V(X) is the number of right cosets of H in G and its valency is |D|/|H|. It follows that the group G in its coset action by right multiplication on V (X) is transitive, and the kernel of this representation of G is the intersection of all the conjugates of H in G. If this kernel is trivial, then we say the subgroup H is core-free. In particular, if H = 1, then we get a Cayley graph. Conversely, each vertex-transitive graph is isomorphic to a coset graph (see [21]). Let G be a group, let L and R be subgroups of G and let D be a union of double cosets of R and L in G, namely, D = |Ji RdjL. By [G : L] and [G : R], we denote the set of right cosets of G relative to L and R, respectively. Define a bipartite graph X = B(G, L, R; D) with bipartition V(X) = [G : L] U [G : R] and edge set E(X) = {{Lg, Rdg} | g G G, d G D}. This graph is called the bicoset graph of G with respect to L, R and D (see [10]). Proposition 2.2. ([10, Lemmas 2.3, 2.4]) (i) The bicoset graph X = B(G, L, R; D) is connected if and only if G is generated by elements of D-1D. (ii) Let Y be a bipartite graph with bipartition V (Y) = U (Y) U W (Y), let G be a subgroup of Aut (Y) acting transitively on both U and W, let u G U (Y) and w G W(Y), and set D = {g G G | wg G Y1(u)}, where Y1(u) is the neighborhood of u. Then D is a union of double cosets of Gw and Gu in G, and Y = B(G, Gu, Gw; D). In particular, if {u, w} G E(Y) and Gu acts transitively on its neighborhood, then D = Gw Gu. Proposition 2.3. ([17, Satz 4.5]) Let H be a subgroup of a group G. Then CG (H) is a normal subgroup of NG(H) and the quotient NG(H)/CG(H) is isomorphic with a subgroup of Aut H. Let G be a group and N a subgroup of G. If there exists a subgroup H of G such that G = NH and N n H = 1, then the subgroup H is called a complement of N in G. The following proposition is due to Gaschutz. Proposition 2.4. ([17, Satz 17.4]) Let G be a finite group. Let A and B be two subgroups of G such that A is abelian normal in G, A < B < G and (|A|, |G : B|) = 1. If A has a complement in B, then A has a complement in G. W. Xu et al.: 2-Arc-Transitive regular covers of Kn,n — nK2... 273 Proposition 2.5. ([7, Lemma 2.7]) Ifp is a prime, then the general linear group GL(2, p) does not contain a nonabelian simple subgroup. A central extension of a group G is a pair (H, n) where H is a group and n : H ^ G is a surjective homomorphism with ker(n) < Z(H). A central extension (G, n) of G is universal if for each central extension (H, a) of G there exists the unique group homomorphism a : G ^ H with n = aa. If G is a perfect group, namely G' = G, then up to isomorphism, G has the unique universal central extension, say (G, n), (see [1, pp.166-167]). In this case, G is called the universal covering group of G and ker(n) the Schur multiplier of G. Proposition 2.6. ([6, page xv]) The Schur multiplier of the simple group PSL(2, q) is Z2 for q = 9, and Z6 for q = 9. The following proposition is quoted from [9]. Proposition 2.7. ([9, Lemma 2.5]) Let Y be a graph and let B be a set of cycles of Y spanning the cycle space CY of Y. If X is a cover of Y given by a voltage assignment f for which each C G B is trivial, then X is disconnected. 3 Proof of Theorem 1.1 Now we prove Theorem 1.1. Let U = {1,2, • • • ,n} and W = {1', 2', • • • ,n'}. Set Y = Kn,n - nK2 (n > 3) with the vertex set V(Y) = U U W and edge set E (Y) = {{i,j'} | i = j,i,j = 1,2 • • • , n}. Let X be a cover of Y with the covering projection $ : X ^ Y and the covering transformation group K = Z^, where p is a prime. Suppose that n = 3. Then Y is a 6-cycle and there is only one cotree arc. Since X is assumed to be connected, all the voltage assigned to the cotree arcs in Y should generate K. It means that K is a cyclic group, a contradiction. Suppose that n = 4. In [12, Theorem 4.1], all regular covers of K4i4 - 4K2 were classified, whose covering transformation group K is either cyclic or elementary abelian, and whose fibre-preserving automorphism group acts arc-transitively. Among them, X (p) is the unique cover when K = Zp and the fibre-preserving automorphism group acts 2-arc-transitively. In what follows, we will assume n > 5. Since our aim is to find the covers of Y whose fibre-preserving automorphism group acts 2-arc-transitively, this group module the covering transformation group K should be isomorphic to a 2-arc-transitive subgroup of Aut Y, in other word, there exists a 2-arc-transitive subgroup of Aut Y to be lifted. Now, let A < Aut Y be a 2-arc-transitive subgroup, and let^ < A be the corresponding index 2 subgroup of A fixing U and W setwise. Let A and G be the respective lifts of A and G. Clearly, Aut (Y) = Sn x (a), where a is the involution exchanging every pair i and i'. Now, we show that G has a faithful 3-transitive representation on the two biparts of Y. Take arbitrary two different triples {ui, v\,w\} and {u2, v2, w2} with ui, vi, wi e U and i e {1,2}. Since (ui,v',wi) and (u2,v2,w2) are both 2-arcs, and since A acts 2-arc-transitively on Y, there exists an element g e A such that (ui,v',wi)g = (u2,v2,w2), noting that v'f = v2 implying vg = v2. Moreover, it is obvious that g fixes two biparts setwise so that g e G. So G acts 3-transitively on U. By the symmetry, G acts 3-transitively on another bipart. Therefore, G should be one of the 3-transitive groups listed in Proposition 2.1. Since n > 5, we conclude the following four cases from Proposition 2.1: 274 Ars Math. Contemp. 10 (2016) 255-268 (1) either soc(G) is 4-transitive or soc(G) = M22; (2) n = 5 and soc(G) = A5; (3) soc(G) = PSL(2, q) with q > 5; (4) G is of affine type, that is the last two cases of Proposition 2.1. To prove the theorem, we shall prove the non-existence for the above four cases separately in the following subsections. 3.1 Either soc(G) is 4-transitive or soc(G) = M22 Lemma 3.1. There exist no regular covers X of Kn,n — nK2, whose fibre-preserving automorphism group acts 2-arc-transitively and whose covering transformation group is isomorphic to Zp with p a prime, such that either soc(G) acts 4-transitively on two biparts or soc(G) = Mp2. Proof. Suppose that G has a nonabelian simple socle T := soc(G) which is either 4-transitive or isomorphic to M22. Let T be the lift of T so that T/K = T. In view of Proposition 2.3, we have (T/K)/(Cf(K)/K) = T/Cf (K) < Aut (K) = GL(2,p). (3.1) Since Cf(K)/K > T/K and T/K is simple, we get Cf(K)/K = 1 or T/K. If the first case happens, then Eq(3.1) implies that GL(2,p) contains a nonabelian simple subgroup, which contradicts Proposition 2.5. Thus, C^,(K) = T, that is, K < Z(T). It was shown in [9, pp.1361-1364] that the voltages on all the 4-cycles and 6-cycles of the base graph Y are trivial, provided K < Z(T) and either T is 4-transitive or T = M22. Therefore, Proposition 2.7 implies that the covering graph X is disconnected. This completes the proof of the lemma. □ 3.2 n = 5 and soc(G) = A5 Lemma 3.2. Suppose that n = 5 and soc(G) = A5. Then, there are no connected graphs X arising as regular covers of Y whose covering transformation group K is isomorphic to Zp with p a prime, and whose fibre-preserving automorphism group acts 2-arc-transitively. Proof. Since G is isomorphic to either A5 or S5, it suffices to consider the case G = A5. Let G be a lift of G, that is, G/K = G. As in Lemma 3.1, a similar argument shows that K < Z(G). Set T := G'. In what follows, we divide our proof into four steps. Step 1: Show T n K =1 or Z2. Set T := G'. Since G' = G, we get Tt/T n K = TtK/K = (G/K)' = G' = G = Gt/K = A5, (3.2) which implies that G = TK. As K < Z(G), we have T = [G, G] = [TK, TK] = [T, T] = T'. W. Xu et al.: 2-Arc-Transitive regular covers of Kn,n — nK2... 275 Thus, T n K < T' n Z(T) and Eq(3.2) implies that T is a proper central extension of T n K by G = A5. By Proposition 2.6, we know that the Schur Multiplier of A5 is Z2. Thus, T n K is either 1 or Z2. Let u G V(Y) be an arbitrary vertex, and take T G ^-1(u), where ^ is the covering projection from X to Y. Step 2: Show D4 < Gs n T Now, we have Ga = G„ = A4 and so Gk/Gk n t 5 G5t/t < G/t = tk/t 5 k/k n t . (3.3) Since Ga n T > Gg = A4, it follows that Ga n T = 1, D4 or A4. If GK n T = 1, then Eq(3) implies that Ga = A4 is isomorphic to a quotient group of K = Zp, a contradiction. So, we get D4 < Ga n T. Step 3: Show T 5 A5 and G = T x K. By Step 1, we know that if n K =1 or Z2. If T n K = Z2, then Eq(3.2) implies that T = SL(2,5) which has the unique involution, contradicting the fact that D4 < Ga n T. Hence, it follows that T n K = 1, and so T = A5 and G = T x K. Step 4: Show the nonexistence of the covering graph X. Suppose that V(Y) = {1, 2, 3, 4, 5}U{1', 2', 3', 4', 5'} and E(Y) = {{i,j'} | i = j, 1 < i,j < 5}. Since T = A5, we may identify T with A5. In T, set x = (23)(45), y = (25)(34), z = (234), b = (15)(23). Then, GF = ((x,y) x (z)) x K, where F = ^-1(1) is the fibre over the vertex 1 G V(Y). Take T G F. Since D4 < Ga n T, one may deduce that D4 = (x, y) < Gjj so that L := Ga = (x, y) x (zk1) for some k1 G K. Note that GF = GF/, where F' = ^_1(1') is the fibre over the vertex 1' g V(Y). Then, one may assume that R := Gw = (x, y) x (zk2) for some k2 G K and w G F'. By Proposition 2.2, the covering graph X should be isomorphic to a bicoset graph X' = B(G, L, R; D), where D = Rbk3L for some k3 G K with two biparts: U = {Lk | k G K}U{Lbx*yjk 1 i, j = 0,1, k G K}, W' = {Rk 1 k G K}U{Rbxy k|i,j =0,1,k G K}. Moreover, X' should satisfy the following two conditions. (i) d(X') = 4: 276 Ars Math. Contemp. 10 (2016) 255-268 Since the length of the orbit of L containing the vertex Rbk3L is 4, zk1 must fix the vertex Rbk3, that is, Rbk3 = Rbk3zk1 = Rbk3zk1(bk3)-1bk3 = Rzb k1bk3 = Rz-1k-1k2kibk3 = Rbk 3k2ki, which implies that k2 = k-1. (3.4) (ii) Connectedness property: By Eq(4), we have (D-1D) = (LbRbL) = (L, Rb) = (x,y,zk1,xb,yb,zbk2)< T x (k1) = G. It follows from Proposition 2.2(i) that the bicoset graph X' is disconnected, which completes our proof. □ 3.3 G is of affine type Lemma 3.3. Suppose that either G = AGL(m, 2), where m > 3 or G = Z| x A7 < AGL(4,2). Then, there are no connected graphs X arising as regular covers of Y whose covering transformation group K is isomorphic to Z'2p with p a prime, and whose fibre-preserving automorphism group acts 2-arc-transitively. Proof. The arguments in both cases are exactly the same, and so here we just discuss the first case in details. Suppose that G = AGL(m, 2) = Zf x GL(m, 2), and let G be a lift of G, namely G/K = G. Since Gd(K)/K > G/K = Zf x GL(m, 2), it follows that Cq (K)/K=l, Zf or G/K. By Proposition 2.3, we have (G/K)/(Cq(K)/K) = G/Cq(K) < Aut (K) = GL(2,p). (3.5) If the first two cases happen, then Eq(3.5) implies that GL(2,p) contains a nonabelian simple subgroup, which contradicts Proposition 2.5. Thus, Cq (K) = G, that is K < Z(G). _ Let A be the group of fibre-preserving automorphism of X acting 2-arc-transitively. Let U and W be the two biparts of X. Take a fibre F in U and take a vertex T1 g F. Set M := GSl = GL(m, 2) and T/Kj= soc(G/K) = Zf. Then G = T x M. Let F' denote the unique corresponding fibre in W without edges leading to F and take a vertex w1 g F'. Then GF = GF>. Since M is the unique subgroup isomorphic to GL(m, 2) in K x M, it follows that GWl = M. First, suppose that p = 2. Now, Gf = K x M. Since (|G : Gf |, |K |) = (2f,p2) = 1, by Proposition 2.4, K has a complement in G. So, we may suppose that G = K x (L x M), where L = Zf. Since G is transitive on W, there exists an element x g G W. Xu et al.: 2-Arc-Transitive regular covers of Kn,n — nK2... 277 such that (wi,wf) G E(X). By Proposition 2.2(ii), X is isomorphic to a bicoset graph B(G,M, Mx; D), where D = MMx. Since L x M > G, we get (D-1D) = (M, Mx) < L x M = G. It follows from Proposition 2.2(i) that X is disconnected. Next, suppose that p = 2, namely K = Z2 x Z2. Let F = {T^ T2,T3,T4} and F' = {wy1, wy2, wy3, wt;4}. Clearly, M has four orbits on U \ F and W \ F', respectively, say Ai, A2, A3, A4; Al, A2, A3, A4. For i = 0,1,2, • • • , by Xi(M1) we denote the set of vertices of distance i from w1. Without loss of generality, let X1 (T ) = A1. Since M acts 2-arc-transitively on the arcs initialed from w1, it follows that X2(w1) is an orbit of M, that is, X2(w1) = Aj for some i G {1, 2,3,4}. Then X3(w1) = {Wj}, for some j G {1, 2, 3,4}. Clearly, X4(w1) = 0 and therefore X is disconnected. □ 3.4 soc(G) = PSL(2, q) for q > 5 In this subsection, identify V(Y) with two copies of the projective line PG(1, q). Lemma 3.4. Suppose that PSL(2, q) < G < PrL(2, q), where q = rl > 5 is an odd prime power. Then, there are no connected graphs X arising as regular covers of Y whose covering transformation group K is isomorphic to Z^ with p a prime, and whose fibre-preserving automorphism group acts 2-arc-transitively. Proof. Let G be the lift of G so that G/K = G. Since PrL(2, q)' = PSL(2, q) and PSL(2, q) < G < PrL(2, q), we have G' = PSL(2, q). Hence, G is insolvable and there exists a positive integer m such that G(m) = C?(m+1). Suppose that T = C?(m), it follows that T/T n K = TTK/K = G?(m)K/K = (G/K)(m) = G(m) = PSL(2, q). (3.6) Therefore, TTK/K is simple and so (TK/K) n (G^j(K)/K) = 1 or TK/K. Again, by Proposition 2.3 and 2.5, we have TK/K < Cg (K)/K, implying that if n K < Z(Tf). Thus, by Eq(3.6), Tf is a proper central extension of Tf n K by PSL(2, q). In viewing of Proposition 2.6, the Schur Multiplier of PSL(2, q) is either Z2 for q = 9 or Z6 for q = 9. It is obvious that T n K_= 1 or Z2 for q = 9. Next, we show it is also true for q =9. Assume, the contrary, that Tf n K = Z3 for q =9. Since TK/K = PSL(2,9), we get (TfK)s = Z3 x Z4. Let = If < (TK)5. As H n K =1 and (|TfK : HfK|, |K|) = 1, it follows from Proposition 2.4 that Khas a complement in TK, say Nf. Thus, TK = K x NT = Z2 x PSL(2, 9). Since [K, T] = 1, one may get Nf = n ' = (:fK)' = [tk, :fK ] = [t, :f] = :f' = t, contradicting T n K = Z3. Therefore we have either T n K = 1 or T n K = Z2. In what follows, we discuss these two cases respectively. Set M := TK so that M/K = PSL(2, q). Case 1: Tfn K = 1 278 Ars Math. Contemp. 10 (2016) 255-268 In this case, we have M = T x K and T = PSL(2, q), and we shall identify T with PSL(2, q). In PSL(2, q), set 1 M x -( ° M y - ° 1 0 1 ) ' x - ^ 0 e-1 ) ' y - ^ -1 ° where F* - (e) and i G Fg. Let Q - {U | i G Fg) = Z[ and Q < T be the lift of Q. Acting on PG(1 , q), set H :- (PSL(2 , q))œ - Q x (x) and the points i G PG(1 , q) \ {to} correspond to the cosets Hytj. Take ù G ^-1(to) and set H :- Mïï. Since H is a lift of H, we may assume that H - Q1 x (xk1) for some k1 G K, and Q1 < Q x K. Actually, we are showing Q1 - Q below. Suppose that Q - Q^ it followsjthat p - r. Then, there exist two nontrivial elements c1 G Q and k G K such that c1k G (Q1. Moreover, we have |¿ù 1 n Q| > r1-2. If l > 2, then there exists a nontrivial element c2 G Q1 n Q. Since (x) has two orbits both with length 2-11 on Q \ {1} by conjugacy action, (xk1) has the same property on Q1 \ {1}, whose jwo orbits should be Bi :- {(cik)} - {c1xfcl>k} and B2 :- {c2xkl>}. Therefore, Q1 - B1 U B2 U {1}. Noting r > 3, the inverse (c1k)-1 of c1k G Q1 is not contained in B1 U B2 U {1}, a contradiction. If l - 1, then we get Q1 n Q - 1. As q - r1 - r > 5, there exist two nontrivial elements C2 G Q and k' G K such that 02k' G QQi. Again, Qi - {c1xkl>k} U {c2xkl>k'} U {1}. Sincep - r > 5, take ks G K\{1, k, k'} for some integer s. Then, (c1k)s - c1sks G Q1 is neither contained in {c1xkl>k} nor in {c2xkl>k'}, a contradiction. If l - 2 and r > 5, we shall have the same discussion as in the case l - 1. Now, we only need to consider l - 2 and r - 3, that is, q - r1 - 9. Since c1k G Q^ it is easy to check that (xk1)-1(c1k)(xk1) - c^k - c-1k G {(c1)k} C (Q1. Hence, 1 - (c1k)(c-1k) - k2 G Q^ a contradiction again. By the above discussion, we may assume that L :- - Q x (xk1) and R :- My -Q x (xk2) for some k1, k2 G K and ù' G ^-1(to'). Then by Proposition 2.2, our graph X is isomorphic to a bicoset graph X' - B(M, L, R; D) for some double coset D with two biparts: | | U - {Lk | k G K} U {Lytjk | i G Fq, k G K}, W' - {Rk 1 k G K} U {Rytjk 1 i G Fq, k G K}. Since there is only one edge from L to the block {Ryk | k G K}, we may assume that the neighbor of L corresponds to the bicoset D - Ryk3L for some k3 G K. Then X' should satisfy the following two conditions. (i) d(X') - q: Since the length of the orbit of L containing the vertex Ryk3 L is q, we have xk1 should fix the vertex Ryk3, that is, Ryk3 - Ryk3xki - Ryk3xki(yk3)-1yk3 - Ryxyk-1kiyk3 - Rx-2k—1kiyk3 - Rk2kiyk3, W. Xu et al.: 2-Arc-Transitive regular covers of Kn,n — nK2... 279 which implies that k2 = k-1. (3.7) (ii) Connectedness property: By Eq(3.7), we have (D-1D) = {L(yk3)-lR(yk3)L) = {L, Ry) _ _ = (Q, xk1, Qy, xyk2) = (Q, xk1, 4, Tran. Amer. Math. Soc. 353 (2001), 3511-3529. [21] P. Lorimer, Vertex-Transitive Graphs: Symmetric Graphs of Prime Valency, J. Graph Theory 8 (1984), 55-68. [22] D. Marusic, On 2-arc-transitivity of Cayley graphs, J. Comb. Theory B 87 (2003), 162-196. [23] C.E. Praeger, An O'Nan-Scott theorem for finite quasiprimitive permutation groups and an application to 2-arc transitive graphs, J. London Math. Soc. 47 (1993), 227-239. [24] C.E. Praeger, On a reduction theorem for finite, bipartite, 2-arc-transitive graphs, Australas J. Combin. 7 (1993), 21-36. [25] W.Q. Xu and S.F. Du, 2-arc-trantive cyclic covers of Kn,n — nK2, J. Algebraic Combin. 39 (2014), 883-902. [26] W.Q. Xu, S.F. Du, J.H. Kwak and M.Y. Xu, 2-arc-transitive metacyclic covers of complete graphs, J. Comb. Theory B 111 (2015), 54-74.