ARS MATHEMATICA CONTEMPORANEA ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 16 (2019) 215-235 https://doi.org/10.26493/1855-3974.1437.6e8 (Also available at http://amc-journal.eu) Edge-transitive bi-p-metacirculants of valency p Yan-Li Qin, Jin-Xin Zhou * Mathematics, Beijing Jiaotong University, Beijing 100044, P.R. China Received 7 July 2017, accepted 23 February 2018, published online 20 December 2018 Let p be an odd prime. A graph is called a bi-p-metacirculant on a metacyclic p-group H if admits a metacyclic p-group H of automorphisms acting semiregularly on its vertices with two orbits. A bi-p-metacirculant on a group H is said to be abelian or non-abelian according to whether or not H is abelian. By the results of Malnic et al. in 2004 and Feng et al. in 2006, we see that up to isomorphism, the Gray graph is the only cubic edge-transitive non-abelian bi-p-metacirculant on a group of order p3. This motivates us to consider the classification of cubic edge-transitive bi-p-metacirculants. Previously, we have proved that a cubic edge-transitive non-abelian bi-p-metacirculant exists if and only if p = 3. In this paper, we give a classification of connected edge-transitive non-abelian bi-p-metacirculants of valency p, and consequently, we complete the classification of connected cubic edge-transitive non-abelian bi-p-metacirculants. Keywords: Bi-p-metacirculant, edge-transitive, inner-abelian p-group. Math. Subj. Class.: 05C25, 20B25 1 Introduction Given a group H, let R, L and S be three subsets of H such that R-1 = R, L-1 = L and RUL does not contain the identity element of H. The bi-Cayley graph over H with respect to the triple (R, L, S), denoted by BiCay(H, R, L, S), is the graph having vertex set the union H0 U H1 of two copies of H, and edges of the form {h0, (xh)0}, {h1, (yh)1} and {h0, (zh)1} with x G R, y G L, z G S and h0 G H0, h1 G H1 representing a given h G H .It is easy to see that a graph is a bi-Cayley graph over a group H if and only if it admits H as a semiregular automorphism group with two orbits. * Supported by the National Natural Science Foundation of China (11671030) and the Fundamental Research Funds for the Central Universities (2015JBM110). E-mail addresses: yanliqin@bjtu.edu.cn (Yan-Li Qin), jxzhou@bjtu.edu.cn (Jin-Xin Zhou) Abstract ©® This work is licensed under https://creativecommons.Org/licenses/by/4.0/ 216 Ars Math. Contemp. 16 (2019) 141-155 Let r = BiCay(H, R, L, S). For g G H, define a permutation R(g) on the vertices of r by the rule hf(g) = (hg)i, Vi G Z2, h G H. Then R(H) = {R(g) | g G H} is a semiregular subgroup of Aut(r) which is isomorphic to H and has H0 and Hi as its two orbits. When R(H) is normal in Aut(r), the bi-Cayley graph r = BiCay(H, R, L, S) is said to be normal (see [24]). When NAut(r) (R(H)) is transitive on the edge set of r, we say that r is normal edge-transitive (see [7]). Bi-Cayley graphs are useful in constructing edge-transitive graphs (see [7, 24]). However, it is difficult in general to decide whether a bi-Cayley graph is edge-transitive. So it is natural to investigate the edge-transitive bi-Cayley graphs over some given groups. Note that metacylic groups are widely used in constructing graphs with some kinds of symmetry, see, for example, [1, 11, 12, 13, 14, 18]. (A group G is called metacyclic if it contains a cyclic normal subgroup N such that G/N is cyclic.) In this paper, we shall focus on the bi-Cayley graphs over a metacyclic p-group with p an odd prime. For convenience, a bi-Cayley graph over a (resp. non-abelian or abelian) metacyclic p-group is simply called a (resp. non-abelian or abelian) bi-p-metacirculant. Note that the Gray graph [6], the smallest cubic semisymmetric graph, is a non-abeian bi-3-metacirculant of order 2 • 33. Malnic et al. in [8,17] gave a classification of cubic edge-transitive graphs of order 2p3 for each prime p. Actually, it is easy to prove that every cubic edge-transitive graphs of order 2p3 is a bi-Cayley graph over a group of order p3. Rather than describe the classification in detail, we would simply like to point out one striking feature: except the Gray graph, there do not exist other cubic edge-transitive non-abelian bi-p-metacirculants of order 2 • p3 for every odd prime p. This seems to suggest that cubic edge-transitive non-abelian bi-p-metacirculants are rare. Motivated by this, we are going to consider the following problem: Problem 1.1. Classify cubic edge-transitive non-abelian bi-p-metacirculants for every odd prime p. In [19], we gave a partial answer to this problem. We first proved that a cubic edge-transitive non-abelian bi-p-metacirculant exists if and only if p = 3, and then we gave a classification of cubic edge-transitive bi-Cayley graphs over an inner-abelian metacyclic p-group for each odd prime p. (A non-abelian group is called an inner-abelian group if all of its proper subgroups are abelian.) In view of this, to solve Problem 1.1, it suffices to classify cubic edge-transitive non-abelian bi-3-metacirculants. Naturally, the following problem arises. Problem 1.2. Classify edge-transitive non-abelian bi-p-metacirculants of valency p for every odd prime p. The following is the main result of this paper which gives a solution of Problem 1.2. Theorem 1.3. Let p be an odd prime, and let r be a connected edge-transitive non-abelian bi-p-metacirculants of valency p. Then p = 3 and r is isomorphic to one of the following graphs: (i) rr = BiCay(Gr, 0, 0, {1, a, a-ib}), Gr = (a, b | a3r+1 = b3r = 1, b-iab = ai+^ ) , Y.-L. Qin andJ.-X. Zhou: Edge-transitive bi-p-metacirculants of valency p 217 (ii) £r = BiCay(Hr, 0, 0, {1, b, b-1a}), Hr = ^a,b | a3r+1 = b3^1 = 1, b-1ab = a1+3^ , where r is a positive integer. Remark 1.4. The graphs rr and £r are actually those graphs what we have found in [19]. By [19], rr is semisymmetric while £r is symmetric. To the best of our knowledge, the graphs rr form the first known infinite family of cubic semisymmetric graphs of order twice a power of 3. From the above theorem and [19, Theorem 1], we may immediately obtain the following result which gives a solution of Problem 1.1. Corollary 1.5. Let p be an odd prime. A connected cubic non-abelian bi-p-metacirculant is edge-transitive if and only if it is isomorphic to one the graphs given in Theorem 1.3. Remark 1.6. The classification of cubic edge-transitive bi-Cayley graphs on abelian groups has been given in [10, 23]. So our result actually completes the classification of all cubic edge-transitive bi-p-metacirculants for each odd prime p. 2 Preliminaries 2.1 Definitions and notation Throughout this paper, groups are assumed to be finite, and graphs are assumed to be finite, connected, simple and undirected. For the group-theoretic and the graph-theoretic terminology not defined here we refer the reader to [4, 21]. Let G be a permutation group on a set Q and take a e Q. The stabilizer Ga of a in G is the subgroup of G fixing the point a. The group G is said to be semiregular on Q if Ga = 1 for every a e Q and regular if G is transitive and semiregular. For a positive integer n, denote by Zn the cyclic group of order n and by Z*n the multiplicative group of Zn consisting of numbers coprime to n. For a finite group G, the full automorphism group and the derived subgroup of G will be denoted by Aut(G) and G', respectively. Denote by exp(G) the exponent of G. For any x e G, denote by o(x) the order of x. For two groups M and N, N x M denotes a semidirect product of N by M .A non-abelian group is called an inner-abelian group if all of its proper subgroups are abelian. For a graph r, we denote by V(r) the set of all vertices of r, by E(r) the set of all edges of r, by A(r) the set of all arcs of r, and by Aut(r) the full automorphism group of r. For u,v e V(r), denote by {u, v} the edge incident to u and v in r. If a subgroup G of Aut(r) acts transitively on V(r), E(r) or A(r), we say that r is G-vertex-transitive, G-edge-transitive or G-arc-transitive, respectively. In the special case when G = Aut(r) we say that r is vertex-transitive, edge-transitive or arc-transitive, respectively. An arc-transitive graph is also called a symmetric graph. A graph r is said to be semisymmetric if r is regular and is edge- but not vertex-transitive. 218 ArsMath. Contemp. 16(2019)203-213 2.2 Quotient graph Let r be a connected graph with an edge-transitive group G of automorphisms and let N be a normal subgroup of G. The quotient graph rN of r relative to N is defined as the graph with vertices the orbits of N on V(T) and with two orbits adjacent if there exists an edge in r between the vertices lying in those two orbits. Below we introduce two propositions of which the first is a result of [15, Theorem 9]. Proposition 2.1. Let p be an odd prime and r be a graph of valency p, andlet G < Aut(T) be arc-transitive on r. Then G is an s-arc-regular subgroup of Aut(T) for some integer s. If N < G has more than two orbits in V(T), then N is semiregular on V(T), rN is a symmetric graph of valency p with G/N as an s-arc-regular subgroup of automorphisms. In view of [16, Lemma 3.2], we have the following proposition. Proposition 2.2. Let p be an odd prime and r be a graph of valency p, andlet G < Aut(T) be transitive on E(T) but intransitive on V(T). Then r is a bipartite graph with two partition sets, say V0 and Vl. If N < G is intransitive on each of V0 and Vi, then N is semiregular on V(T), rN is a graph of valency p with G/N as an edge- but not vertex-transitive group ofautomorphisms. 2.3 Bi-Cayley graphs Proposition 2.3 ([23, Lemma 3.1]). Let r = BiCay(H, R, L, S) be a connected bi-Cayley graph over a group H. Then the following hold: (1) H is generated by Ru£u S. (2) Up to graph isomorphism, S can be chosen to contain the identity of H. (3) For any automorphism a of H, BiCay(H, R, L, S) = BiCay(H, Ra, La, Sa). (4) BiCay(H, R, L, S) = BiCay(H, L, R, S-i). Let r = BiCay(H, R, L, S) be a bi-Cayley graph over a group H. Recall that for each g e H, R(g) is a permutation on V(r) defined by the rule hf{9) = (hg)i, Vi e z2,h,g e H, and R(H) = {R(g) | g e H} < Aut(r). For an automorphism a of H and x,y,g e H, define two permutations on V(r) = H0 U Hl as following: Sa,x,y: ho ^ (xha)i, hi ^ (yha)o, Vh e H, Va,9: h0 ^ (ha)0, hL ^ (gha)i, Vh e H. Set I = {6aiXy I a e Aut(H) s.t. Ra = x-iLx, La = y-iRy, Sa = y-1S-ix}, F = { n, we have „m pm pm (xy)p = xp yp , Vx, y G G. (2) For any positive integer k and for any x,y G G, xpk = ypk ^ (x-1y)pk = 1 ^ (xy-1)pk = 1. Proof. By [22, Theorem 2.1], it suffices to prove the items (1) and (2). Since G' is cyclic, (1) follows from [9, Chapter 3, §10, Theorem 10.2 (c) and Theorem 10.8 (g)]. Item (2) follows from [9, Chapter 3, § 10, Theorem 10.2 (c) and Theorem 10.6 (a)]. □ Lemma 3.2. Let p be an odd prime, and let H be a metacyclic p-group generated by a, b with the following defining relations: aPm = bPn = 1, b-1ab = a1+Pr, where m, n, r are positive integers such that r < m < n + r. Then the following hold: (1) For any i G Zpm, j g Zpn, we have aib = bai(1+pr)j. (2) For any positive integer k and for any i G Zpm, j g Zpn, we have (Vai)k = bkjai ^ks=0(1+pr)sj. 2t (3) For any positive integers t, k and any element x of H, if xp = 1, then x(1+pt)k = x1+k-pt. (4) The subgroup of H of order p is one of the following groups: (ar-) , (bpn-1 ai'pm-1) (i' G Zp). 220 ArsMath. Contemp. 16(2019)203-213 Proof. From [19, Lemma 14 (1) -(2)], we have the items (1) - (2). For (3), the result is clearly true if k = 1. In what follows, assume k > 2. Since xp = 1, we have xp = 1. Then = xc0'(pt)° • xc1(Pt)1 • xc2-(p4)2 • • • xck•(pt)k = x • (xpt)C1 • (xP2t)C2 ••• (xPkt)Ck — x • xkpt = x1+kpt, and so (3) holds. (Here for any integers N > l > 0, we denote by C'N the binomial coefficient, that is, C'N = n^N-iy..) For (4), let Q1(H) = (x G H | o(x) = p}. Since H is ametacyclic p-group, by [2, Exercise 85], we have Q1(H) = Zp x Zp. It implies that H has p +1 subgroups of order p. Furthermore, the subgroup of H of order p is one of the following groups: (a,Pm-1) , (bP"-1 ai Pm-1) (i' G Zp), as required. □ 4 Inner-abelian bi-p-metacirculants of valency p In this section, we focus on edge-transitive bi-Cayley graphs over inner-abelian metacyclic p-groups of valency p. For convenience, a bi-Cayley graph over an inner-abelian metacyclic p-group is simply called an inner-abelian bi-p-metacirculant. In [19, Theorem 2], we gave a classification of cubic edge-transitive inner-abelian bi-p-metacirculants. Proposition 4.1 ([19, Theorem 2]). Let r be a connected cubic edge-transitive bi-Cayley graph over an inner-abelian metacyclic 3-group H. Then H = Gr or Hr, and r = rr or Er, where the groups Gr, Hr, and the graphs rr, Er are defined as in Theorem 1.3. In particular, H/H' = Z3r X Z3r or Z3r x Z3r + 1 . In this section, we shall prove the following theorem. Theorem 4.2. Let H be an inner-abelian metacyclic p-group with p an odd prime, and let r be a connected edge-transitive bi-Cayley graph over H of valency p. Then p = 3, and r is isomorphic to one of the graphs given in Theorem 1.3. 4.1 Two technical lemmas Lemma 4.3. Let p be an odd prime and let r be a connected edge-transitive graph of valency p. If G < Aut(r) is transitive on the edges of r, then for each v G V(r), |Gv | = pm with (m,p) = 1. Proof. Since G is transitive on the edges of r, for each v G V(r), the order of a vertex stabilizer Gv must be divisible by p. Suppose, by way of contradiction, that | Gv | is divisible by p2. Let G*v be the subgroup of Gv fixing the neighborhood r(v) of v in r pointwise. (1+Pt) - ,Jc°-ifc-(pt)°+ci-ifc-1-(pt)1+c2-ifc-2-(pt)2H—hck•i°^(pt)fc] x =x Y.-L. Qin andJ.-X. Zhou: Edge-transitive bi-p-metacirculants of valency p 221 Then Gv/G*v < Sp, forcing that p | \G*v |. Then G* contains an element a of order p. Note that each orbit of (a) has length either 1 or p. Since (a) fixes v and each vertex in r(v), the connectedness of r implies that each orbit of (a) has length 1, and so a = 1, a contradiction. □ Lemma 4.4. Let H be a p-group with p an odd prime, and let r = BiCay(H, R, L, S) be a connected edge-transitive bi-Cayley graph of valency p. Then (1) r is normal edge-transitive, R = L = 0, and S = {1, h, hha,..., hha ■ ■ ■ h°P 2} for some 1 = h G H and a G Aut(H) satisfying hhaha ■ ■ ■ haP = 1 and o(a) \ p; (2) if H has a characteristic subgroup K such that H/K is isomorphic to zpm x Zpn, then \m — n| < 1. Proof. Let A = Aut(r), and let P be a sylow p-subgroup of A such that R(H) < P. Since r is edge-transitive, Lemma 4.3 gives that |A| = |R(H)|^ p ■ m, where (p, m) = 1. It follows that |P| = p|R(H)|, and hence P < Na(R(H)). Furthermore, for any e G E(r), we have |A : Ae| = |E(r)| = p|R(H )|,andso |Ae| = m. It follows that Pe = P nAe = 1, and hence |P : Pe| = |P| = p|R(H)| = |E(r)|. Thus, P is transitive on the edges of r. Thus, r is normal edge-transitive. Let N = Na(R(H)). Then N is transitive on the edges of r. Since R(H) < N, the two orbits H0, Hi of R(H) do not contain any edge of r, and so R = L = 0. By Proposition 2.3, we may assume that 1 G S. Since N is transitive on the edges of r and r has valency p, Nlo has an element of order p for some a G Aut(H) and 1 = h G H. Furthermore, cyclically permutes the elements in r(10). So we have r(10) = {1i,hi, (hha)i,..., (hha ■■■ haP-2 )i} and hhah"2 ■■■ haP-1 = 1. This implies that P2 S = {1,h,hha,...,hha ■■■ ha }, and haP = h. Since r is connected, one has H = (S) = (ha% | 0 < i < p — 1). As h°P = h, ap is a trivial automorphism of H. Consequently, we have o(a) = 1 or p and (1) is proved. For (2), without loss of generality, assume that H/K = zpm x zpn with m > n, where K is a characteristic subgroup of H. Let T = (R(x) G R(H) | xpn G K). Then T is characteristic in R(H) and R(H)/T = Zpm-n. Propositions 2.1 and 2.2 implies that the quotient graph rT of r relative to T is a graph of valency p with N/T as an edge-transitive group of automorphisms. Clearly, R(H)/T is semiregular on V(rT) with two orbits and R(H)/T < N/T, so rT is a normal edge-transitive bi-Cayley graph over R(H)/T = Zpm-n of valency p. So to complete the proof, it suffices to show that if H = Zpm then m < 1. Suppose to the contrary that H = Zpm with m > 2. Since H = (ha | 0 < i < p — 1), we have H = (h). Let ha = hx for some A G z*pm. Then 1 = hhaha2 ■ ■ ■ h°P-1 = h1+x+x2+ -+XP-1, and then 1 + A + A2 + ■■■ + Ap-1 = 0 (mod pm). 222 Ars Math. Contemp. 16(2019)203-213 It follows that Ap = 1 (mod pm), and hence A = 1 (mod p). Let A = kp +1 for some integer k. Since m > 2, we have 1 + (kp + 1) + (kp + 1)2 + • • • + (kp + 1)p-1 = 0 (mod p2). It follows that 1 + (kp + 1) + (2kp + 1) +-----+ ((p - 1)kp + 1) = 0 (mod p2), and hence p + ^p(p — 1)kp = 0 (mod p2). A contradiction occurs. □ 4.2 Proof of Thorem 4.2 Throughout this subsection, we shall always let H be an inner-abelian metacyclic p-group with p an odd prime, and r be a connected edge-transitive bi-Cayley graph over H of valency p. In view of Lemma 4.4(1) and since H is inner abelian, we may make the following assumption throughout this subsection. Assumption 4.5. r = BiCay(H, 0, 0, S), where S = {1, h, hha,..., hha • • • haP 2} for some 1 = h G H and a G Aut(H) satisfying hhaha • • • haP = 1 and o(a) = p. Proof of Theorem 4.2. Suppose to the contrary that p > 3. Since H is an inner-abelian metacyclic p-group, by elementary group theory (see also [20] or [3, Lemma 65.2]), we may assume that H = ^a, b | apt+1 = bP' = 1,b-1ab = , where t > 1, s > 1. Note that H/H' = (aH') x (bH') = zpt x zps. By Lemma 4.4, we have H/H' = (aH') x (bH') = Zpt x Zpt, Zpt x Zpt+i or Zpt x Zpt-i. If H/H' = (aH') x (bH') = Zpt x Zpt-i, then s = t - 1 and H = ^a, b | apt+1 = bpt- = 1,b-1ab = . Let T = (R(x) | x G H,xpt-1 = 1). Then T is characteristic in R(H) and R(H)/T is isomorphic to Zp2. However, by the proof of Lemma 4.4, this is impossible. If H/H' = (aH') x (bH') = zpt x Zpt, then s = t and H = ( a, b I ap ^a, b | aPt+1 = bp = 1,b-1ab = apt+1) where t > 1. We shall show that this is impossible in Lemma 4.6. If H/H' = (aH') x (bH') = Zpt x Zpt+1, then s = t +1 and H = (a, b | aPt+1 = bPt+1 = 1, b-1ab = a^1) , where t > 1. We shall show that this is impossible in Lemma 4.7. □ Y.-L. Qin andJ.-X. Zhou: Edge-transitive bi-p-metacirculants of valency p 223 Lemma 4.6. If H = (a, b | apt+l = bpt = 1, b-1ab = apt+1^ (t > 0), then p = 3. Proof. Suppose to the contrary that p > 3. We first define the following four maps. Let Y : a ^ a1+p, b ^ b, S: a ^ a,b ^ b1+p, a: a ^ a,b ^ bap, t : a ^ ba, b ^ b. Let x1 = a1+p, x2 = x3 = a, x4 = ba, y1 = y4 = b, y2 = b1+p and y3 = bap. Since H is an inner-abelian metacyclic-p group, by Proposition 3.1 and a direct computation, we have o(xi1) = o(a) = pt+1, o(yil) = o(b) = pt and it is direct to check that xil and yil have the same relations as do a and b, where i1 G {1,2, 3,4}. Moreover, for any i1 G {1, 2, 3,4}, we have (xil, yil} = H. It follows that each of the above four maps induces an automorphism of H. Set P = (a, y, S, t}. By a direct computation, we have o(y) = pt, o(S) = pt-1 and o(a) = o(t) = pt. Furthermore, yS = Sy, Y-1aY = ap+1 and S-1aS = ae with l(p +1) = 1 (mod pt). As both y and S fixes the subgroup (b} while a does not, one has (a, y, S} = (a} x ((y} x (S}) = Zpt x (Zpt x Zpt-l). Observing that (a, y, S} fixes the subgroup (a} setwise but t does not, it follows that (a, y, S} n (t} = 1, and hence |P| > p4t-1. In view of [13, Theorem 2.8], Aut(H) has a normal Sylow p-subgroup of order p4t-1. It follows that P = (a, y, S, t} is the unique Sylow p-subgroup of Aut(H). In particular, we have P = (y}(S}(a}(T}. Recall that S = {1, h, hha, ...,hha ■ ■ ■ haP-2}. Assume that h = buav for some u G Zpt and v G Zpt+l. Since H = (S}, we have o(h) = exp(H). It follows that (v,p) = 1. Then the map p1: a ^ av,b ^ b induces an automorphism of H. Let p = (tu^1 )-1. Then p G Aut(H) and hv = a. By Proposition 2.4(3), we have that r = F = BiCay(H, 0, 0,SLet P = p-1ap. Then aPa G Aut(r') cyclically permutates the elements in r'(10). It follows that S^ = {1, a, aaP, aaPaP", .. ., aaPaP" ■ ■ ■ app-2}, and aapaP ■ ■ ■ app l = 1. Clearly, o(P) = o(a) = p, so P G P. We assume that P = yiSjaktl for some i,k,l G Zpt and j G Zpt-l. By Lemma 3.2(2)-(3) and Proposition 3.l(l), we have P: | b ^ (b ■ (bla)pk)(1+p)j = b(1+p)j(1+pfci)a(1+p)jpk (4.1) Let U1(H) = {xp | x G H}. Then ^ (H) < Z(H) and a bla ■ w P: U b / (4.2) \bh^b ■ w' for some w, w' G u1(H). Since r' is connected, by Proposition 2.3, we have H = By Proposition 3.1(1), it follows that (l,p) = 1. We shall finish the proof by the following steps. 224 ArsMath. Contemp. 16(2019)203-213 Step 1: t > 1. Suppose to the contrary that t = 1. Then H = (a, b | ap = bp = 1, b-1ab = a1+p). We shall first show that for any r > 1, a8" = brl a1+1 r(r-1)klP+irP (4.3) By Equation (4.1) we have ja ^ bla1+ip ' |b ^ bakp So Equation (4.3) holds when r =1. Now assume that r > 1 and ar-1 = b(r-1)'a1+1 (r-1)(r-2)fcip+j(r-1)p. By a direct computation, we have = (b(r-1)la1+ 2 (r-1)(r-2)fcip+i(r-1)p)8 a = (bafcP)(r-1)'(b'a1+ip)1+ 2 (r-1)(r-2)fcip+i(r-1)p = b(r-1)l a(r-1)lkp b'a1+1 i(r-1)2-(r-1)]fcip+irp = b(r-1)l+la1+[ 2 (r-1)2- 1 (r-1) + (r-1)]fcip+irp _ bri„1+1 r(r-1)klp+irp By induction, we have Equation (4.3). Now we show that for any r > 1, a • a8 • • • = b2r(r+1)la(r+1) + [6r(r+1)(2r+1)i+ 2r(r+1)i+ 6(r-1)r(r + 1)fci]p. (4.4) By Equation (4.3) and Lemma 3.2(1)&(3), we have a • a8 = a • bla1+iP = bla(1+P)!a1+iP = bla1+lPa1+iP = bla2+(l+i)P So Equation 4.4 holds when r = 1. Now assume that r > 1 and a • a8 • • • a8r-1 = b2 (r-1)War+[6 (r-1)r(2r-1)l+ 2 (r-1)ri+1 (r-2)(r-1)rfci]p. By a direct computation, we have 8 82 8r aa8a8 •• • a8 = b 1 (r-1)rlar+[ 1 (r-1)r(2r-1)i +1 (r-1)ri+1 (r-2)(r-1)rfci]p _ bWa1 +1 r(r-1)fcip+irp = b 1 r(r+1)l a{r+[ 6 (r-1)r(2r-1)l+1 (r-1)ri+ 6 (r-2)(r-1)rfci]p}-(1+rip) + 1 + 2 r(r-1)fcip+irp = b 1 r (r+1) l ar (1+rlp) + [ 6 (r-1)r(2r-1)i+ 2 (r-1)ri+1 (r-2)(r-1)rfci]p+1+ 2 r(r-1)fcip+irp = b2r(r+1)la(r+1)+[ 1 (r-1)r(2r-1)+r2]lp+[2 (r-1)r+r]ip+[6 (r-2)(r-1)+ 2r(r-1)]rfcip = b2r(r+1)la(r+1)+[6r(r+1)(2r+1)l+ 2r(r + 1)i+ 6 (r-1)r(r+1)fci]p By induction, we have Equation (4.4). Y.-L. Qin andJ.-X. Zhou: Edge-transitive bi-p-metacirculants of valency p 225 Since p is a prime and p > 3, by Equation (4.4), we have «aßaß2 • • • aßP-1 = b 1 (P-^V+11 (p-i)p(2p-i)i+1 (p-i)pi+ 6(p-2)(p-1)pfci]p = «p = ^ a contradiction. Step 2: A final contradiction Let 02(H) = {xp2 | x € H}. Then Ö2(H) < Z(H). By Equation (4.1), we have a1+ip • to, bß = b1+jp+pklapk • to', for some to, to' € Ö2(H). Let m = i/ (mod p), n = i (mod p), f = j + k/ (mod p) for some m, n, f € Zp. Then fa ^ bmp+lanp+1 • to1 (45) ß: |b ^ bfP+1akP • to1 (.5) for some to1,to1 € Ö2(H). We shall first prove the following claim. Claim. For any r > 2, aßr = bCrP+2IadrPTOr for some cr, dr € Zp and TOr € Ö2(H). Since t > 1, for any positive integer i0, by Lemma 3.2(1)&(3), we have abi0 = bi0a(1+pt)i0 = bi0a1+iopt = bioa • to0, (4.6) for some to0 € Ö2(H). Then by Equations (4.5) and (4.6), we have aß2 = (bfp+1akp • to 1 )mp+l(bmp+l«np+1 • TO1)np+1 • to£ = b(2m+/i+ni)p+2ia(2n+fci)p ^ for some to2 € Ö2(H). Take c2, d2 € Zp such that 2m + f/ + n/ = c2 (mod p) and 2n + k/ = d2 (mod p). If r = 2, then Claim is clearly true. Now assume that r > 2 and Claim holds for any positive integer less than r. Then for some cr_1, dr-1 G Zp and TOr_ ; G Ö2(H), and then aßr = (6fp+1afcp • to;)1 p+2l(bmp+lanp+1 • TOi)dr-lp • to£ r_ 1 = b(cr-i + 2/i+idr-i)p+2ia(2fci+dr_i)p _ for some TOr G Ö2(H). Take cr, dr G Zp such that cr-1 + 2/1 + 1dr-1 = cr (mod p) and 2k1 + dr-1 = dr (mod p). By induction, we complete the proof of Claim. Now by our Claim, we have /p = bcPp+2Iadpp • TOp for some cp, G Zp and top g H2(H). It follows that cpp + 21 = 0 (mod p2), a contradiction. This completes the proof of our lemma. □ r 226 Ars Math. Contemp. 16 (2019) 141-155 Lemma 4.7. If H = (a,b | apt+l = bpt+l = 1, b-1ab = apt+1^ (t > 0), then p = 3. Proof. Suppose to the contrary that p > 3. We first define the following four maps. Let Y : a ^ a1+p, b ^ b, 6: a ^ a,b ^ b1+p, a : a ^ bpa, b ^ b, t : a ^ a,b ^ ba. Let x1 = a1+p, x2 = x4 = a, x3 = bpa, y1 = y3 = b, y2 = b1+p and y4 = ba. Since H is an inner-abelian metacyclic-p group, by Proposition 3.1 and a direct computation, we have o(xil) = o(a) = pt+1, o(yil) = o(b) = pf' and it is direct to check that xil and yil have the same relations as do a and b, where i1 G {1,2, 3,4}. Moreover, for any i1 G {1, 2, 3,4}, we have (xil,yil} = H. It follows that each of the above four maps induces an automorphism of H. Set P = (a, y, 6, t}. By a direct computation, we have o(y) = o(6) = pl, o(a) = pf' and o(t) = pt+1. Moreover, we have y6 = 6y, S-1aS = ap+1 and Y-1aY = ae with £(p +1) = 1 (mod pi+1). As both y and 6 fixes the subgroup (a} while a does not, one has (a,Y,6} = (a} x ((y} x (6}) = Zpt x (Zpt x Zpt). Observing that (a, y, 6} fixes the subgroup (b} setwise but t does not, it follows that (a,Y, 6} n (t} = 1, and hence |P| > p4i+1. In view of [13, Theorem 2.8], Aut(H) has a normal Sylow p-subgroup of order p4i+1. It follows that P = (a, y, 6, t} is the unique Sylow p-subgroup of Aut(H). In particular, we have P = (Y}(6}(a}(T}. Recall that S = {1, h, hha, ...,hha ■■■ haP-2} and o(a) = p. Assume that h = buav for some u G Zpt+l and v G zpt+l. Since H = (S}, we obtain that o(h) = exp(H). It follows that (u,p) = 1. Then there exists u' G Z*t+l such that u = u'v (mod pi+1). Let p = au(6u)-1 (tv)-1. Then p G Aut(H) and h* = b. By Proposition 2.4(3), we have r = BiCay(H, 0, 0, S*). Let r' = BiCay(H, 0, 0, S*) and 3 = p-1ap. Then G Aut(r') cyclically permutates the elements in r'(10). It follows that bbf3 b$ ■■■ b$p =1 and s* = {1, b, bb$, bb$ b$2,..., bb$ b$2 ■ ■ ■ b$p-2}. Since o(3) = o(a) = p, we have 3 G P. Assume that 3 = Yi6jakt 1 for some i,j,k G zpt and l G zpt+l. Then by Lemma 3.2(2)-(3) and Proposition 3.1(1), we have fa ^ (ba1 )(1+p)ikpa(1+p)i = b^*^ a(1+p)i(1+klp) 3 : U ^ (ba1)(1+p)j = b(1+p)ja(1+p)j 1 (4.7) and then ß : ? ^ l\ Wt (4.8) \b^bal ■ w' for some w, w' G u1(H). Since r' = r is connected, we derive from Proposition 2.3 that H = (Sv). By Proposition 3.1(1), it follows that (l,p) = 1. We shall finish the proof by the following steps. Y.-L. Qin andJ.-X. Zhou: Edge-transitive bi-p-metacirculants of valency p 227 Step 1: t > 1. Suppose to the contrary that t = 1. Then H = (a, b | ap = bp =1,b-1ab = a1+p). We shall first show that for any r > 1, = b1+(rJ +1 r(r-1)fci)pari+ 2 r(r+1)jp+ 2 r(r-1)(i+fci)ip+1 r(r-1)(r-2)fci2p. (4.9) By Equation (4.7), we have ja H bfcpa1+(i+fcl)p ' jb H b1+jpal+jlp. Thus Equation (4.9) holds when r =1. Now assume that r > 1 and = b1 + ((r-1)j+ 2 (r-1)(r-2)fci)p . a(r-1)l+ 2(r-1)rjlp+ 2(r-1)(r-2)(i+fci)ip+1 (r-1)(r-2)(r-3)fci2p By a direct computation, we have b^" = (b1+jPa'+j'P)1 + ((r-1)j+ 2 (r-1)(r-2)fci)p • (bfcPa1+(i+fc')P)(r-1)'+ 2 (r-1)rjlp+ 2 (r-1)(r-2)(i+fci)ip+ 6 (r-1)(r-2)(r-3)fci2p = b1+(rJ +1 (r-1)(r-2)fci + (r-1)fci)p _ ai+jp+[(r-1)ij + 2 (r-1)(r-2)fci2]p . a(r-1)i(1+(i+fci)p)+1 (r-1)rjp+ 2 (r-1)(r-2)(i+fci)ip+ 6 (r-1)(r-2)(r-3)fci2p = b1+rJP+[ 1 (r-1)(r-2) + (r-1)]fcip . a[i+(r-1)i] + [1+(r-1)+1 (r-1)r]jp+1 r(r-1)(i+fci)ip+[ 1 +1 (r-3)](r-1)(r-2)fci2p = b1+(rJ +1 r(r-1)fci)pari+ 2 r(r+1)jlp+1 r(r-1)(i+fci)ip+ 6 r(r-1)(r-2)fci2p By induction, we have Equation (4.9). Then by Equation (4.9), we have = b1 + (pj+ 2 p(p-1)fc')pap'+ 2 p(p+1)jlp+1 p(p-1)(i+fci)ip+ 6 p(p-1)(p-2)fci2p = bapl = b, a contradiction. Step 2: A final contradiction. 1 2 Let ^2(H) = {xp | x € H}. Then ^(H) < Z(H). By Equation (4.7), we have • w b^ = bjp+1ajlp+l • w for some w, w' € u2(H). Let f = i + kl (mod p), n = j (mod p), m = jl (mod p) for some m, n, f € Zp. Then fa H bfcpafp+1 • w' ^' H b"p+1a™p+(4.10) for some w1,w1 € H2(H). We shall first prove the following claim. 228 ArsMath. Contemp. 16(2019)203-213 Claim. For any r > 1, b^ = brnp+ ^^ kip+iarmp+ i^v1' (n+f)iP+r(r ^ 2) ki2P+ri. ^ with wr € H2(H). If r = 1, then by Equation (4.10), Claim is clearly true. Now assume that r > 1 and Claim holds for any positive integer less than r. Then b?r-1 = b(r-1)np+ (r-1)2(r-2) klp+1 , a(r-1)mp+ (r-1)2(r-2) (n+f )lp+ (r-1)(r.2)(r-3) kl2p+(r-1)l , ^ for some wr-1 € U2(H). Since t > 1, for any positive integer i0, by Lemma 3.2(1)&(3), we have abio = bioa(1+pt)i0 = bioa1+iopt = bioa • ro0, (4.11) for some w0 € U2 (H). Then by Equations (4.10) and (4.11), we have (r 1)(r 2) , = (b"p+1amp+l • ro1)(r-1)"p+ ^-2 -) klp+1 afp+1 • )(r-1)mp+ (r-12(r-2) (n+f)lp+ (r-1)(r.2)(r-3) kl2p+(r-1)l ^ ^ b(r-1)np+ (r-1)2(r-2) klp+np+1+k(r-1)lp _ ^ _ a(r-1)nlp+ (r-1)2(r-2) kl2p i amp+l + (r-1)mp+ (r-1)2(r-2) (n+f )lp+ (r-1)(r.2)(r-3) kl2p+(r-1)l + (r-1)f lp = b™p+ r(r2-1) klp+1 ^ armp+ r(r2-1) (n+f )lp+ r(r-16(r-2) kl2p+rl ^ ^ for some wr € U2 (H). By induction, we complete the proof of Claim. Now by our Claim and o(P) = p, we have = bnp2 + ^ klp2 + 1 ^ amp2 + ^ (n+f )lp2+ (p-1)6(p-2) kl2p2+pl ^ w = b for some € H2(H). It follows thatpl = 0 (mod p2), a contradiction. This completes the proof of our lemma. □ 5 Proof of Theorem 1.3 We first prove a lemma. Lemma 5.1. Let p be an odd prime, and let H be a metacyclic p-group. If r is a connected edge-transitive bi-Cayley graph over H of valency p, then H is either abelian or inner-abelian. Proof. We may assume that H is non-abelian. By Proposition 3.1, the group H has the following presentation: H = (a, b | apr+s+" = 1, bpr+s+t = apr+s, ab = a^ ) , where r, s, t, u are non-negative integers with u < r and r > 1. Let r = BiCay(H, R, L, S) be a connected edge-transitive bi-p-Cayley graph over H of valency p. Let A = Aut(r), and let P be a Sylow p-subgroup of A such that Y.-L. Qin andJ.-X. Zhou: Edge-transitive bi-p-metacirculants of valency p 229 R(H) < P. From the proof of Lemma 4.4(1), we see that P is transitive on the edges of r. Since H' = (apr} = Zps+u, we have H/H' = ( a, b | op = b ^ i,ab — - Zpr X Zpr+s + t , where a = aH' and b = bH'. By Lemma 4.4(2), we have s + t = 0 or 1, and so (s,t) = (0,0), (1,0) or (0,1). Let n = 2r + 2s+u+1. We use induction on n. If n = 1 or 2, then H is clearly abelian, as desired. Assume n > 3. Let N be a minimal normal subgroup of P and N < R(H). Since H is metacyclic, we have N = Zp or Zp x Zp. Suppose that N = Zp x Zp. Note that R(H)' = Zps+u. Let Q be the subgroup of R(H)' of order p. Since Q is characteristic in r(h)' and R(H)' is characteristic in R(H), R(H)

1. If r = 1, then by Magma [5], there is no cubic edge-transitive bi-Cayley graph over H, a contradiction. If r > 2, then by Lemma 4.4(1), we have R = L = 0. Assume that S = {1, g, h}. Since r is connected, by Proposition 2.3(1), we have H = (S} = (g, h}. It follows that o(g) = o(h) = exp(H) = 3r+2, and so H' = (x3r} = (g3r} = (h3r}. Moreover, by Lemma 4.4(1), there exists a G Aut(H) such that ga = g-1h, ha = g-1 and o(a) | 3. Suppose that a is trivial. Then h = g-1, and then H = (g}, a contradiction. Thus, a has order 3. Assume that (g3r )a = gA• 3r for some A G Zg. Then (h3r )a = hA •3r. Since ga = g-1h and ha = g-1, we have gA •3r = g-3r h3r and hA •3r = g-3r. Then gA2 = (gA •3r ) A = (g-3r h3r ) A = g-A•3r hA •3r = g-A •3rg-3r = g(-A-1) • 3r . It follows that g(A2+A+1) • 3r = 1, and so 9 | A2 + A + 1, a contradiction. Case 2: (s,t) = (1,0). In this case, we have TT /71 3r+u+1 ,3r+1 3r+1 b 1+3r\ H =( a, b | a3 =1,b3 = a3 , « = a > 1 Y.-L. Qin andJ.-X. Zhou: Edge-transitive bi-p-metacirculants of valency p 231 Let x = a and y = ba 1. Since b3 = a3 , by Proposition 3.1(2), we obtain that y3r+1 = (ba-1)3r+1 = 1 and xy = aba-1 = (ab)a-1 = (a1+3r )a-1 = a1+3^ = x1+3^. Then R(H) = H = (x,y | x3r+"+1 = y3r+1 = 1,xy = x1+3^ , Recall that N = Z3 and N < R(H). By Lemma 3.2(4), N is one of the following four groups: (x3 ), (y3 ), (y3 x3 ), (y3 x2 3 ). Suppose first that N = (x3 ). Then x has order 3r+"+1. We shall show that H/N has the following presentation: H/N = (x,h | x3r+u+1 = h3r =T,xh = x1+3 Actually, if N = (y3r ), then we may take h = y. If N = (y3rx3r+" ), then take h = yx3", and then by Lemma 3.2(2)- (3), we have / 3u\3r : (yx ) = y 3" x3u [1+(1 3r•(3r-1) = y3r x3u[1+(1+3r ) + (1+2 • 3r) + • • • + (1+(3r-1) • 3r )] = y3"X3"[3r + 3 2 • 3r] or q " + r = y3x3 € N. If N = (y3"x2 • 3"+"}, then take h = yx2• 3", and then by Lemma 3.2(2)-(3), we have (yX2• 3" )3r = y3rx2 • 3"[1 + (1+3r) + (1+3r)2 + • • • + (1+3r)3r-1] = y3r x2 • 3"[1 + (1+3r ) + (1+2 • 3r) + • • • + (1 + (3r-1) • 3r)] = y3r x2 • 3" [3r + 3"-(32r-1) • 3r] = y3" x2 • 3"+r € N. Clearly, in each case, we have xh = x1+3". So H/N always has the above presentation. Since R(H)/N is inner-abelian, by [20] or [3, Lemma 65.2], we have u = 0. Then 3r+1 3r+1 - y H = (x,y | x3r+1 = y3r+1 =1,xy = x1+3^ , where r > 1. By [20] or [3, Lemma 65.2], H is inner-abelian, as required. Suppose now N = (x3r+"). Then R(H)/N = (x, y | x3r+" = y3r+1 =1,xy = x1+3^) . Since R(H)/N is inner-abelian, by [20] or [3, Lemma 65.2], we have u = 1. Then H = (x, y | x3r+2 = y3r+1 =1,x« = x1+3^) , 232 ArsMath. Contemp. 16(2019)203-213 where r > 1. If r = 1, then by Magma [5], there is no cubic edge-transitive bi-Cayley graph over H, a contradiction. If r > 2, then by Lemma 4.4(1), we have R = L = 0. Assume that S = {1, g, h}. Since r is connected, by Proposition 2.3(1), we have H = (S} = (g, h}. It follows that o(g) = o(h) = exp(H) = 3r+2. By Lemma 4.4(1), there exists a e Aut(H) such that ga = g-1h, ha = g-1 and o(a) | 3. Suppose that a is trivial. Then h = g-1, and then H = (g), a contradiction. Thus, a has order 3. Note that nr(H) = ^z3" | z e H^ = (x3") x ^y3"^ = Z9 x Z3 and g3" ,h3" e ^ (H). If (g3"} = (h3"}, then we may assume that (g3" )a = gA '3" for some A e Zg. Then (h3")a = hA '3". Since ga = g-1h and ha = g-1, we have gx3" = g-3"h3" and hA'3" = g-3". Then gA2- 3" = (gA • 3" )A = (g-3" h3" )A = g-A • 3" hA • 3" = g-A • 3" g-3" = g(-A-1) • 3". It follows that g(A2+A+1) • 3" = 1, and so 9 | A2 + A + 1, a contradiction. Suppose (g3"} = (h3"}. Then Qr(H) = (g3", h3"} and H' = (x3"} = Z9. Assume that x3" = gl•3" hj •3" for some i, j e Z9. Then either (i, 3) = 1 or (j, 3) = 1. Since H' = (x3"}, we have (x3"}a = (x3"}. So (g4 •3"hj •3")a = (g4 •3"hj •3")k for some k e Z9. Then gik •3" hjk •3" = (g4 •3" hj •3" )a = (ga)4 •3" (ha)j •3" = g-i •3" V 3" g-j •3" = g-(i+j) • 3" h •3". It follows that — (i + j) = ik (mod 9) and i = jk (mod 9). Then — (jk + j) = jk2 (mod 9), and so j (1 + k + k2) = 0 (mod 9), forcing that 3 | j. Furthermore, since i = jk (mod 9), we have 3 | i, a contradiction. Case 3: (s,t) = (0,1). In this case, we have H = (a,b | a3"+" = 1,b3^ = a3", ab = a1+3" ) . Let x = b, y = b3a-1. Since ab = a1+3", we have b-1 aba-1 = a3", and then aba-1 = ba3" = bb3"+1 = b1+3"+1. Since b3" = a3", by Proposition 3.1(2), we have x3"+"+1 = b3"+"+1 = a3"+" = 1, y3" = (b3a-1)3" = 1, xy = bb3a-1 = (b)a-1 = aba-1 = b1+3"+1 = x1+3"+1. Then R(H) = H = (x, y | x3"+"+1 = y3" = 1, x" = x1+3"+^ . Recall that N = Z3 and N < R(H). By Lemma 3.2(4), N is one of the following four groups: (x3"+"}, (y3" 1}, (y3" 1 x3"+"}, (y3" 1 x2 • 3"+"}. Y.-L. Qin andJ.-X. Zhou: Edge-transitive bi-p-metacirculants of valency p 233 Suppose first that N = {x3 ). Then x has order 3r+"+1. We shall show that H/N has the following presentation: H/N = /x,h | x3r+u+1 = h3'-1 = 1, xh = x1+3r+1 Actually, if N = (y3r }, then we may take h = y. If N = (y3r x3 "}, then take h = yx3"+1, and then by Lemma 3.2(2) -(3), we have (yx3" + 1 )3r-1 = y3r-1 x3u + 1[1 + (1+3r+1) + (1+3r+1)2H-----h(1+3r+1)3r-1-1] = y3r-1 x3u + 1 [1 + (1+3r+1 ) + (1+2-3r+1)H-----+(1 + (3r-1-1)-3r+1)] or —1 ou + 1 To r — 1 i 3r 1-(3r 1 — 1) or+1] = y3 x3 [3 + 2 3 ) or—1 ou+r = y3 x3 + G N. If N = (y3r 1 x23r+u}, then take h = yx2 3u+1, and then by Lemma 3.2(2)-(3), we have (yx23U + 1 )3r —1 = y3r —1 x2-3u + 1[1+(1+3r+1) + (1+3r+1)2 + --- + (1+3r+1)3r —1 —1] = y3r —1 x2*3u + 1 [1+(1+3r+1) + (1+2-3r+1 )+-----+(1+(3r —1 — 1) • 3r+1 )] = y3r —1 x2 3u + 1[3r —1 + 3r —1-(32r—1 —1) •3r+1] r 1 u+r = y3 x2 3 + G N. Clearly, in each case, we have xh = x1+3r. So H/N always has the above presentation. Since R(H)/N is inner-abelian, by [20] or [3, Lemma 65.2], we have u = 1. However, by Proposition 4.1, there is no cubic edge-transitive bi-Cayley graph over R(H)/N, a contradiction. Suppose now that N = (x3r+u}. Then R(H)/N = (x,y | x3r+u = y3r = T,x¥ = x1+3r+1 ) . Since R(H)/N is inner-abelian, by [20] or [3, Lemma 65.2], we have u = 2. However, by Proposition 4.1, there is no cubic edge-transitive bi-Cayley graph over R(H)/N, a contradiction. □ Now we are ready to finish the proof of Theorem 1.3. Proof of Theorem 1.3. By Lemma 5.1, if H is non-abelian, then H is inner-abelian. By Theorem 4.2, we have p = 3, and then by Proposition 4.1, r is isomorphic to either rr or £r, as desired. □ References [1] B. Alspach and T. D. Parsons, A construction for vertex-transitive graphs, Canad. J. Math. 34 (1982), 307-318, doi:10.4153/cjm-1982-020-8. [2] Y. Berkovich, Groups of Prime Power Order, Volume 1, volume 46 of De Gruyter Expositions in Mathematics, De Gruyter, Kammergericht, Berlin, 2008, doi:10.1515/9783110208221. 234 Ars Math. Contemp. 16(2019)215-235 [3] Y. 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