¿^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 7 (2014) 41-54 A class of semisymmetric graphs Li Wang School of Mathematical Sciences, Capital Normal University, Beijing, 100048, PR China and School ofMathematics and Information Sciences, Henan Polytechnic University, Jiaozuo, 454000, P R China Shaofei Du * School of Mathematical Sciences, Capital Normal University, Beijing, 100048, PR China Xuewen Li Department ofMathematics and Information Sciences, Tangshan Normal University, Tangshan, 063000, P R China Received 20 November 2011, accepted 1 October 2012, published online 14 January 2013 Abstract A simple undirected graph is said to be semisymmetric if it is regular and edge-transitive but not vertex-transitive. Every semisymmetric graph is a bipartite graph with two parts of equal size. Let p be a prime. In this paper, a class of semisymmetric graphs of order 2p3 are determined. This work is a partial result for our long term goal to classify all semisymmetric graphs of order 2p3. Keywords: Permutation group, primitive group, vertex-transitive graph, semisymmetric graph. Math. Subj. Class.: 05C10, 05C25, 20B25 1 Introduction All graphs considered in this paper are finite, undirected, connected and simple. For a graph X, we use V(X), E(X) and A := Aut(X) to denote its vertex set, edge set and the full automorphism group, respectively. The graph is said to be vertex-transitive and edge-transitive, if A acts transitively on V(X) and E(X), respectively. If X is bipartite with bipartition V(X) = W(X) U U(X), we let A+ be the subgroup of A preserving both * Corresponding author. E-mail addresses: wanglimath@hpu.edu.cn (Li Wang), dushf@mail.cnu.edu.cn (Shaofei Du), lixuewen@sina.com (Xuewen Li) ©® This work is licensed under http://creativecommons.Org/licenses/by/3.0/ 42 Ars Math. Contemp. 7 (2014) 41-54 W(X) and U(X). Since X is connected, we have that either |A : A+| = 2 or A = A+, depending on whether or not there exists an automorphism which interchanges the two parts. For G < A+, the graph X is said to be G-semitransitive if G acts transitively on both W(X) and U(X), and semitransitive if X is A+-semitransitive. We call a graph semisymmetric if it is regular and edge-transitive but not vertex-transitive. It is easy to see that every semisymmetric graph is a semitransitive bipartite graph with two parts of equal size. The first person who studied semisymmetric graphs was Folkman. In 1967 he constructed several infinite families of such graphs and proposed eight open problems see [13]. Afterwards, Bouwer, Titov, Klin, I.V. Ivanov, A.A. Ivanov and others did much work on semisymmetric graphs see [2, 4, 16, 17, 18, 30]. They gave new constructions of such graphs and nearly solved all of Folkman's open problems. In particular, Iofinova and Ivanov [16] in 1985 classified biprimitive semisymmetric cubic graphs using group-theoretical methods. This was the first classification theorem for such graphs. More recently, following some deep results in group theory which depend on the classification of finite simple groups and some methods from graph coverings, some new results on semisymmetric graphs have appeared. For instance, in [11] Du and Xu classified semisymmetric graphs of order 2pq for two different primes p and q. For more papers on semisymmetric graphs see [5, 7, 8, 9, 10, 11, 12, 19, 21, 22, 23, 24, 25, 26, 27, 28, 33]. In [13], Folkman proved that there are no semisymmetric graphs of order 2p and 2p2 where p is a prime. Then we are interesting in determining semisymmetric graphs of order 2p3, wherep is prime. Since the smallest semisymmetric graphs have the order 20 [13], we let p > 3. It is proved in [25] that the Gray graph of order 54 is the only cubic semisymmetric graph of order 2p3. To classify all semisymmetric graphs of order 2p3 is still one of attractive and difficult problems. These graphs X are naturally divided into two subclasses: (1) Aut(X) acts unfaithfully on at least one part; (2) Aut(X) acts faithfully on both parts. Now we are going to concentrate on Subclass (1). To state our main theorem, we first introduce two concepts. Let Y be a connected semitransitive and edge-transitive graph with bipartition V (Y) = W (Y)U U (Y),where W (Y) = Zp and U (Y) = Zp for an odd prime p. For distinguishing the vertices of W (Y) and U (Y) convenience, the vertices of W (Y) and U (Y) are denoted by (i, j, k, 0) and (y, z, 1), respectively, where i, j, k, y, z G Zp. Now we define a bipartite graph X with bipartition W(X) U U(X), where W(X) = W(Y), U(X) = Zp x U(Y) = {(x, y, z, 1) | x, y, z G Zp}, such that two vertices (i, j, k, 0) G W(X) and (x, y, z, 1) G U(X) are adjacent if {(i, j, k, 0), (y, z, 1)} G E(Y). From now on, we shall say that the graph X is the graph expanded from Y and that the graph Y is the graph contracted from X. Clearly X is edge-transitive and regular. Furthermore, since for any (y, z, 1) G U(Y), the p vertices {(x, y, z, 1) | x G Zp} in U(X) have the same neighborhood, X is semisymmetric, provided there exist no two vertices in W (X) which have the same neighborhood. Clearly, Aut(X) acts unfaithfully on W(X) and Aut(X)/Sp = Aut(Y). Note that the semisymmetric graphs where two vertices have the same neighbourhood have been studied in several papers see [11, 20, 29, 33], with different definitions, for L. Wang, S. Du and X. Li: A class of semisymmetric graphs 43 instance, X is a derived graph from Y, X is a unworthy graph, X is contracted to from Y and so on. Let Y be a connected graph and B an imprimitive system of Aut(Y). Define a graph Z with the vertex set B such that two blocks are adjacent in Z if there exists at least one edge in Y between two blocks. This graph Z is called the block graph of Y. Moreover, if B is the set of orbits of some nontrival normal subgroup N of Aut(Y), then we call Z the block graph induced by N. The following proposition gives a characterization for Subclass (1) given in [31]: Proposition 1.1. Suppose X is a semisymmetric graph of order 2p3, where p is an odd prime, such that Aut(X) acts unfaithfully on at least one part. Then Aut(X) must act unfaithfully on one part and faithfully on the other part, and X is the graph expanded from the graph Y with bipartition V(Y) = W(Y)UU(Y), where W(Y) = Zp and U(Y) = Zp,. Moreover, we have that either (1) p = 3, Aut(Y) = S3 I S3 which acts primitively on W (Y); or (2) Aut(Y) has blocks of length p2 on W(Y) and of length p on U(Y). Let Y be the block graph of Y. Then either (2.1) the block graph Y is of valency at least 3, and Aut(Y) is solvable and contains a normal regular subgroup on W(Y); or (2.2) the block graph Y is of valency 2, where Aut(Y) may be solvable or insolvable. Following Proposition 1.1, in this paper we shall determine the graphs in Case (2.2), while Cases (1) and (2.1) will be determined in our another paper. Before giving the main theorem of this paper, we first define six families of graphs Y. Definition 1.2. We shall define six families of bipartite graphs X with bipartition V(X) = W(X) U U(X), where W(X) = {(i,j,k, 0) I i,j,k e Zp}, U(X) = {(x,y,z, 1) | x,y,z e Zp}, and edge set E (X) = {{(i,j,k, 0), (x,i + b,k + , 1)} ^,j,k,x e Zp,b e E} U {{(i,j,k, o), (x, j + sb,k + p++1, 1)} I i,j,k,x e Zp,b e £}, p — 1 where s = 6, Zp = (0) for the family of graphs X2(p, r), s =1 for other five families of graphs Xj(p, r), and E is given by (1) Graphs Xi(p, r): Let p > 3 and let E be a subgroup of Zp of order r, where (p,r) = (7, 3), (11,5). Moreover, the valency of X1 (p,r) is 2pr and the smallest examples are X1(3,1) and X1(3, 2). (2) Graphs X2(p, r): Let p > 5 and let E be a subgroup of Zp of order r > 2, where (p, r) = (7,3), (11, 5) and 2r | (p — 1). Moreover, the valency of X2(p, r) is 2pr and the smallest example is X2 (5,2). (3) Graphs Xs(11,5): Let p =11 and let E = {0, 2,3,4,8} c Zii. Moreover, the valency of X3(11, 5) is 110. 44 Ars Math. Contemp. 7 (2014) 41-54 (4) Graphs X^ll, 6): Let p = 11 and £ = {1,5,6, 7, 9,10} c Zn. Moreover, the valency of X4 (11, 6) is 132. (5) Graphs X5(p, r): Choose a point (v) and a hyperplane L in the project space PG(n - 1, q), where q^irf = p > 7, and let G = (t) be a Singer subgroup of PGL(n, q). Let £ = {l G Zp | (v) G L4'}, where r = |£| = . Moreover, the valency of X5(p, r) is 2pq and the smallest example is X5(7, 3). (6) Graphs X6(p, r) : Adopting the same notation as in (5), set £ = {l G Zp | (v) G L4'}, where r = qn-1. Moreover, the valency of X6(p, r) is 2pqn-1 and the smallest example is X6(7,4). Remark 1.3. For 1 < i < 6, let Xj(p, r) be as in Definition 1.2. Then (1) For any given y, z G Zp, the p vertices {(x, y, z, 1) | x G Zp} have the same neighborhood. Let Yj(p, r) be the contracted graph from Xj(p, r), obtained by contracting each such p vertices into one vertex while preserving the adjacent relation, that is, W(Yi(p,r)) = W(Xi(p,r)), U(Yi(p,r)) = {(y,z, 1) | y, z G Zp}. Then we shall see from the proof of Theorem 1.4 that Aut(Yi(p, r)) = K x D2p, where the subgroup K is the following (i) Yi(p,r) and Yi(p,r): K = Sp if r G {1,p - 1}; K = (Zp x Zr)p if r G {1,p -1}; (ii) Ys(11, 5) and Yt(11,6): K = (PSL(2,11))p; (iii) Y5(p, r) and Y6(p,r): K = (PrL(n,q))p. (2) For any k G Zp, let Wk(Y) = {(i, j,k, 0) G W(Y) | i, j G Zp}, Uz(Y) = {(y, z, 1) | y G Zp}. Then we shall see from the proof of Theorem 1.4 that {Wk (Y) | k G Zp} and {Uz (Y) | z G Zp} are orbits of the group K on W (Y) and U (Y), respectively. Let Y be the block graph induced by K. Then Y is a cycle of length 2p. Now we give the main theorem of this paper. Theorem 1.4. For an odd prime p, suppose that X is a semisymmetric graph of order 2p3 expanded from a graph Y such that Aut(Y) has the blocks of length p2 on W (Y) and of length p on U (Y) while the block graph Y is a cycle of length 2p. Then X is isomorphic to one of graphs Xj(p, r) where 1 < i < 6, defined in Definition 1.2. After this introductory section, some preliminary results will be given in Section 2, and the main theorem will be proved in Sections 3. For group-theoretic concepts and notation not defined here the reader is refereed to [6, 15]. L. Wang, S. Du and X. Li: A class of semisymmetric graphs 45 2 Preliminaries First we introduce some notation. By H char G, we mean that H is a characteristic subgroup of G. Given a group G and a subgroup H of G, by Cos(G, H) we denote the set of right cosets of H in G. The action of G on Cos(G, H) is always assumed to be the right multiplication action. For two subgroups N < G and H < G, by N x H we denote the semi-direct product of N by H, where N is normal. For a group G, by Exp (G) we denote the least common multiple of orders of all the elements of G. By H i K, we denote the wreath product of H and K. A group-theoretic construction of semitransitive and semisymmetric graphs were given in [11]. Here we quote one definition and two results. Definition 2.1. 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. Define a bipartite graph X = (G, L, R; D) with bipartition V(X) = Cos(G, L) U Cos(G, R) and edge set E(X) = {(Lg, Rdg) | g e G, d e D}. This graph is called the bi-coset graph of G with respect to L, R and D. Proposition 2.2. [11] The graph X = B(G, L, R; D) is a well-defined bipartite graph. Under the right multiplication action of G on V(X), the graph X is G-semitransitive. The kernel of the action of G on V(X) is CoreG(L) n CoreG(R), the intersection of the cores of the subgroups L and R in G. Furthermore, we have (i) X is G-edge-transitive if and only if D = RdL for some d e G; (ii) the degree of any vertex in Cos(G, L) (resp. Cos(G, R)) is equal to the number of right cosets of R (resp. L) in D (resp. D-1), so X is regular if and only if |L| = |R|; (iii) X is connected if and only if G is generated by elements of D-1D; (vi) X = B(G, L°, Rb; D') where D' = Ui Rb (b-1dia)La, for any a, b e G; (v) X = B(G, LCT, RCT; DCT) where a is an isomorphism from G to G (it does not appear in [11] but it is easy to prove.) Proposition 2.3. [11] Suppose Y is a G-semitransitive graph with bipartition V (Y) = U(Y) U W(Y). Take u e U(Y) and w e W(Y). Set D = {g e G | wg e Yi(u)}. Then D is a union of double cosets of Gw and Gu in G, and Y = B(G, Gu, Gw; D). Proposition 2.4. [32, 11.6, 11.7] Every permutation group of prime degree p is either insolvable and 2-transitive, or isomorphic to Zp x Zs for some s dividing p — 1. Proposition 2.5. [14] The insolvable permutation groups of prime degree p are given as follows, where T denotes be the socle of the group and H denotes a point stabilizer of T: (i) T = Ap and H = Ap-i; (ii) T = PSL(n, q) and H is the stabilizer of a projective point or a hyperplane in PG(n — 1, q), and |T : H| = (qn — 1)/(q — 1) = p; (iii) T = PSL(2,11) and H = A5, and T has two conjugacy classes of subgroups isomorphic to A5; (iv) T = M11 and H = M10; 46 Ars Math. Contemp. 7 (2014) 41-54 (v) T = M23 and H = M22. Lemma 2.6. [31] Let G be an imprimitive transitive group of degree p2 with p > 3 and p3 | |G|. Suppose that G has an imprimitive system B with p-blocks and the kernel K. Let P be a Sylow p-subgroup of G and N = P n K. Then (1) Exp (P) < p2, |Z(P)| = p andP = N(t), where tp e Z(P); (2) K is solvable, N char K and so N < G, provided either p = 3; or p > 5 and |N| < pp-1. 3 Proof of the main theorem To prove Theorem 1.4, we assume that p is an odd prime and that X is a semisymmetric graph of order 2p3 expanded from the graph Y, where Aut(Y) acts edge transitively on Y and has blocks of length p2 on W(Y) and of length p on U(Y), and the block graph Y is a cycle C2p of length 2p. Let F = Aut(Y) and let B = {Bo,Bi, ••• ,Bp_i} and B' = {B'0 ,B[, ••• ,B'p_ 1} be blocks system of F on U (Y) and W (Y), respectively. Label E(Y) = {(bo, B'p+1), (B'p+1 , bi), • • • , (bp_i , B'0), (B'0, Bp+1), • • • , (bp_i, b'p_), (B'p_i , bo)}, 22 2 2 22 so that Y = C2p. Set /p — 1 p + 1 ~2 a =(0,1, ••• ,p - 1) and t = (0)(1, -1) ••• (^- , ) G Sp. Then Aut(Y) ^ (a,r) ^ D2p, by defining (Bj)7 = Bi-, and (Bj)Y = BjT for any 7 G (a, t). Label the vertices in Bi by a^ for j G Zp. By considering the imprimitive action of F on U (Y), we know that F < Sp I (a,T) = Sp x (a,T), where, for any e = (e(0),e(1), ••• ,e(p-1)) G Sp and 7 G (a, t ), we have ,(e;Y) = a ... i = V« iY . In particular, by identifying (1,7) with 7 so that aY = a^, we have that (a, t) can be viewed as a subgroup of F. From now on, for any t e T < Sp and i e Zp, we set i+i ti = (mT-~~,t, 1, ••• , 1) and Tj = (t, 1 t e T), where Tj acts transitively on B, and fixes Bj pointwise for all j = i. Moreover, we have J-(e;7) __T^CT* _ rp _ rp T^a* _ n tj = jY , T0 = = Ti, B 0 = Bi. a Since Kis a transitive group of degree p, following Propositions 2.4 and 2.5 we need to consider the following four cases separately in four subsections: L. Wang, S. Du and X. Li: A class of semisymmetric graphs 47 (i) p > 5 and KBi is insolvable; (ii) p > 5 and KBi = Zp x Zr for r = 1; (iii) p > 5 and KBi = Zp. (iv) p = 3. 3.1 KBi is insolvable for p > 5 Lemma 3.1. Suppose that p > 5 and KBi is insolvable. Then Y is isomorphic to one of the following graphs: (i) Yi(p, r), and Aut(Y) = Sp I D2p, where r = 1 or p — 1; (ii) Y3(11,5) and Y4(11,6), and Aut(Y) = PSL(2,11) I D22; (iii) Ys(p, ^_) and Ye(p, qn-1), and Aut(Y) = PrL(n, q) \ D2P. Proof. Suppose that p > 5 and KBi is insolvable. Then by Lemma 2.6 we know that K = To x T1 x • • • x Tp-1, where T is an insolvable group of degree p and Ti is defined as before. In particular, a Sylow p-subgroup of F is of order pp+1, and so we may assume that F contains a defined as above. Let u g B0 and take an element g0 G Fu \ K. Since g0 fixes B0 setwise and exchanges B'p-i and Bp+i, there exists a d = (d(0), d(1), • • • , d(p_1)) G Sp such that g0 = dr, where 2 2 P t is defined as before. Since F/K = D2p, by considering the order of F we get F = KR where R = (a, dT}. Let Ho = (To)u. Then Ku = H0 x T1 x • • • x Tp-1 and Fu = Ku x (dT}. By Kf = Ku, we know that d(0) G N{sp)0 (Ho) and d(i) G N(sp)4 (T<) for i = 0. Now dT fixes the block B'0 setwise and exchanges Bp-i and Bp+i. Take w G B'0. Since Tp-i xTp+i fixes u and acts transitively on B0, there exists a k G Tp-i xTp+i < Ku 2 2 0 2 2 such that kdT fixes both u and w, where without loss of generality, we denote kd by d again so that dT fixes both u and w. Then Kw = T0 x • • • x Tp-3 x L p—i x Np+i x Tp+3 x • • • x T„ 1 and Fw = Kw (dT}. 2 2 2 2 L By KWT = Kw, we know that L = N and d(i) G N(Sp). (Li) for i G {pT1, p+1}. Now the corresponding groups H and L are two maximal subgroups of T of index p. Following Proposition 2.5 we need to consider three cases separately. (1) H and L are conjugate in T. Without loss of generality, let H = L. For any almost simple group T in Sp, its point stabilizers have two orbits in each block Bi with the respective length 1 and p — 1. We may therefore let T = Sp so that H = Sp-1 and F = SpR = Sp(a, dT} = Sp(a, t}. Thus, we may set d = 1. For later use, we set t = (0,1, • • • ,p — 1), E1 = {0} and S2 = Zp. (2) soc(T) = PSL(2,11), and H and L are not conjugate in T. 48 Ars Math. Contemp. 7 (2014) 41-54 In this case T = PSL(2,11), and T has two nonequivalent representations on the set of right cosets of A5 of cardinality 11. Now F = Tp(a,dr). Since d(i) G NSl1 (T) = T», we have d G Tp and so F = Tp(a, t). Therefore, we set d = 1. Moreover, T may be considered as the automorphism group of a (11, 5, 2)-design D. Let V = Zu be the point set and let t = (0,1, • • • , 10) be an element of order 11 in T. Then M = {0, 2, 3,4,8} c V is a block (see [1, p.55]) of D. Without loss of generality, we choose L and H to be the stabilizes of the block M and point 0, respectively. Again, for later use, set S3 = M and S4 = Zn \ M. (3) soc(T) = PSL(n, q), and H and L are not conjugate in T. In this case, PSL(n, q) < T < Nsp(PSL(n, q))=PrL(n, q). With the same reason as (1), we let T = NSp (PSL(n, q)) and d =1. Let Si and S2 be the set of points and hyperplanes ofPG(n,q - 1), respectively, where |S1| = |S2| = ^—i1 = p. Without loss of generality, we choose L and H to be the stabilizers of a given point (v) and a hyperplane L, respectively. Let G = (t) = Zp be a singer subgroup of PGL(n, q). Let £5 = {l G Zp | (v) G £i!}, £e = Zp \ Si = {l G Zp | (v) G L }, where IS5I = ^q-r1 and |S6| = qn-1. Now for the above three cases (1)-(3), we have Cos(F,Fw) = {Fwti-i tjp+i | i,j,k G Zp}, Cos(F,F„) = {F„ty| y, z G Zp}. 2 ~2 Clearly, Fw has two orbits on Bp-i U Bp+i, that is, 2 2 D = {Fut0aV,Fut0a^ | b G Si}, where l = 1, 3,4, 5,6. For any point Fwtp— tjp+i in W(Y), since _ i p+i p+i _ i _ i F„t0atUtjp+i = F„t0(t 5. As for the graph with the edge set E1, let ^ be a map on W (Y) U U (Y) which fixes W (Y) pointwise and sends (y, z, 1) to (y +1, z, 1). Then ^ is an isomorphism between the present graph and Y1(p, 1). From the proof we know that both Y1(p, 1) and Y1(p,p - 1) have the automorphism group Sp I D2p. □ 3.2 KBi ^ Zp x Zr for p > 5 and r = 1 Lemma 3.2. Suppose KBi = Zp x Zr forp > 5 andr = 1. Then Y = Y1(p,r) or Y2(p,r) where p > 5, r = 1, p - 1 and (p, r) = (7,3), (11, 5), where Aut(Y) = (Zp x Zr) I D2p. Proof. Step 1: Determination of the structure of F. Proof. Suppose KBi = Zp x Zr for r = 1. Let S = (t) x (c) = Zp x Zp-1 < Sp. Then p— 1 we may set T = (t) x (h), where h = c. Let P be a Sylow p-subgroup of F and take d0a € P where do € (to) x (t1) x • • • x (tp_1) = Zp. Then K < Tp and F = K (d0a, dr) for some d € Sp. Moreover, F < F = Tp (d0a, dr) = Tp (a, dr) and (a, dr) / (Tp n (a, dr)) = D2p. Let w € B0 and (B0,Bp—), (B0,Bp+i) € E(Y). Let (w,u1) € E(Y) for « € Bp-i. Then E = (w,u1)F < (w, u1)F. Since the orbits of Fw and Fw on the block Bp-i in U(Y) are completely the same, we have |(w,u1)F| = 2rp3 = |E|, which implies E = (w, u1)F. Therefore, we may just consider the case F = F = Tp(a, dr). As in the last Lemma, we choose two vertices u € B0 and w € B'0 which are fixed by dr. Without loss of generality, let H = (h) so that Fu = (H0 xT1 x • • • xTp-1)(dr), Fw = (T0 xT1 x-xHxHp+i x • • • xTp-1)(dr). We then need to determine the element d. Let d = (d(0), d(1), • • • , d(p-1)) € Sp. Since dr normalizes K, Ku and Kw, it follows that d(i) € Nsp(Hi) = (c) for i € {0, ^}, and d(i) € Nsp(T) = S = (t)(c) for i € {0, p±±1}. Suppose that i € {0, p±±1} and write d(i) = tmcn. Since T < Fu and Fw we may re-choose d(i) = cn. Therefore, for any i € Zp, we get d(i) € (c). (1) Since (dr)2 € K, we have drdr = ((d(0))2, d(1)d(p-1), • • • , d(p-1)d(1)) € K, and by taking into account (1) we get (d(0))2,d(1)d(p-1), ••• ,d( ) d( ^) € H. (2) 50 Ars Math. Contemp. 7 (2014) 41-54 p— i p+1 Since Kw fixes only one point ua 2 in Bp—i and ua 2 in Bp+i and since dr normalizes 2 2 Kw and exchanges Bp—i and Bp+i , it follows that dr must exchange these two points. 2 2 Therefore, F> ^ (dT ) = F„(d( "ï1 ),d( "+1 ), ••• ,d(p-1),d(°),d(1), ••• ,d( "ï3 )}ra = F„d(d( "ï1 ),d( "ï3 ), ••• ,d(1),d(°),d(P-1), ••• ,d( "+1 ) )a = F„(d(°)d("ï1 ), d(Dd("ï3), • • • , d("ï1 )d(°), d()d(P-D, ••• ,d(P-!)d( V ^ p+1 = F„a 2 . Hence, d(°)d(^>, d(1)d(3>, • • • , d(p—1>d(°>, d(^ )d(p-1), • • • , d(p-1)d(^) e H. (3) From (2) and (3) we get d(0),d(1) • • • , d(p-1) e H, or d(0),d(1) • • • , d(p-1) e (cp2—1 > \ H if 2r I (p - 1). (4) Therefore, if 2r f (p - 1) then we set d = 1; if 2r | (p - 1), we set d = (c'm, c'm, • • • c'm) p — 1 where c' = cand m = 0,1. To unify these two cases, in the first case we still write d = (c'm, c'm, • • • c'm) for m = 0. Suppose that 2r | (p - 1). Let F1 = K x (a, r> and F2 = K x (a, dr>, where p— 1 d = (c', • • • , c') with c' = c, noting that c' e T. we may then state the following fact Fact: F1 = F2 Proof: Assume the contrary. Suppose that 7 is an isomorphism from F1 to F2. Since ((t, t, • • • , t)> is characteristic in F1 and F2, we get Y((t,t••• t)) = (tk,tk,tk ••• ,tk) for some k e Fp. Assume that y(t) = edr, where e = (e(0), e(1) • • • , e(p-1)) e K. Since t-1(t,t, ••• ,t)T = (t,t, ••• ,t). we have that is which implies y(t-1b((M, ••• ,t))7 (t ) = y(M, ••• ,t), (edT )-1(tk ,tk, ••• ,tk )(edT ) = (tk ,tk, ••• ,tk ), (tk )e^c' = tk. Therefore, e(0)c' G (t) and so c' G T, a contradiction. Step 2: Determination of the bicoset graphs. Set D(l) = F„t'aand by Z = Z(p, r, d, l) we denote the corresponding bicoset graph. We consider two cases separately. L. Wang, S. Du and X. Li: A class of semisymmetric graphs 51 (1) l = 0. Since Fua= Fuaand Fua^ Kw (dr) = FuaP+21, we have p — 1 - p—1 P+1 - D(0) = Fua — Fw = {Fua —, Fua}. For any point Fwt®p_i tjp+1 ak in W(Z), since 2 2 _ i p+1 p+1 _ I _ I Fuap21 tjp+i_ Fu(tU y(tl4 )jafc+ ^ = Fut0t{ak+ ^ = Fut0 2 2 2 2 and similarly F n^ ti tj nk _ F tj k+^ Fu 5 Lemma 3.3. The case KB = Zp cannot occur. Proof. Suppose that KB = Zp. Then |E(Y)| = 2p3. As above, let w e B'0 and (B0, Bp-i), (B0, Bp+i) e E(Y). Let (w,u1) e E(Y) for uL e Bp-i. Then E = (w,u1)F. We may consider the group F = Sp x (a, t} > F. From the proof of Lemma 3.1, we may construct two representations of F with respective degree p3 and p2 such that both Kw and (Sp)w fix u1. Then (w, u1)F c (w,u1)F. Since |(w,u1)F| = 2p3 = |(w, u1)F|, we have (w, u1)F = (w, u1)F = E(Y) and so Aut(Y) = F, contrary to our hypothesis KBi = Zp. Therefore, this case cannot occur. □ 3.4 p = 3 Lemma 3.4. If p = 3, then Y = Yl(3, r) for r =1,2. Proof. In this case, take F = S3 I D6 and H = L = Z2. Checking the proof of Lemma 3.1(1), one may find that the arguments in there still hold for p = 3. 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