Arc-transitive Dihedrants of Odd Prime- power Order

Arc-transitive Dihedrants of Odd Primepower Order

István Kovács

Graphs and Combinatorics ISSN 0911-0119 Volume 29 Number 3 Graphs and Combinatorics (2013) 29:569-583 DOI 10.1007/s00373-012-1134-6

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Author's personal copy Graphs and Combinatorics (2013) 29:569–583 DOI 10.1007/s00373-012-1134-6 ORIGINAL PAPER

Arc-transitive Dihedrants of Odd Prime-power Order István Kovács

Received: 23 May 2010 / Revised: 15 January 2012 / Published online: 14 Febuary 2012 © Springer 2012

Abstract Let G be a finite group with identity element 1, and S be a subset of G such that 1 ∈ / S and S = S −1 . The Cayley graph Cay(G, S) has vertex set G, and x, y in G are adjacent if and only if x y −1 ∈ S. In this paper we classify the connected, arc-transitive Cayley graphs Cay(D2 pn , S), where D2 pn is the dihedral group of order 2 p n , p is an odd prime. Keywords

Cayley graph · Arc-transitive graph · Dihedral group

1 Introduction All groups in this paper are finite, and all graphs are finite, simple and undirected. For a graph let V (), E() and Aut() denote the vertex set, the edge set and the group of all automorphisms of , respectively. A k-arc of is a sequence of k + 1 vertices v1 , v2 , . . . , vk+1 in V (), not necessarily all distinct, such that any two consecutive terms are adjacent and any three consecutive terms are distinct. We say that is k-arc-transitive if Aut() acts transitively on the set of k-arcs of . Usually 1-arc-transitive graphs are also called arc-transitive. Let G be a group with identity element 1, and S ⊆ G with 1 ∈ / S. The Cayley digraph Cay(G, S) over G relative to S is the graph with vertex set G, and (x, y) in G × G is an arc if and only if x y −1 ∈ S. If in addition S = S −1 = {x −1 | x ∈ S}, then (x, y) is an arc of Cay(G, S) if and only if (y, x) is also, thus Cay(G, S) is usually regarded as an undirected graph, and

I. Kovács supported in part by “ARRS—Agencija za raziskovanje Republike Slovenije”, program no. P1-0285. I. Kovács (B) UP IAM and UP FAMNIT, University of Primorska, Muzejski trg 2, 6000 Koper, Slovenia e-mail: [email protected]

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the term Cayley graph is used. The question that which Cayley digraphs Cay(G, S) are k-arc-transitive for some k > 0 has been studied extensively. In this direction all 2-arc-transitive Cayley digraphs are classified over abelian groups in [12], and over dihedral groups in a sequence of papers [3,13,14]. As for arc-transitivity, a complete classification is known over cyclic groups (see [5,10,15]). In this paper we consider the class of connected, arc-transitive Cayley graphs over dihedral groups, in other words, the class of connected, arc-transitive dihedrants. Notice that, every connected, arc-transitive Cayley digraph over a dihedral group is necessarily undirected. Such graphs have been studied in a number of papers under various additional restrictions (see e.g. [7–9,17,18]). Here we make a further step toward a complete classification of this class, and determine the graphs which have order 2 p n , p is an odd prime. Our result can be viewed as an analog to the classification of arc-transitive circulants (Cayley digraphs over cyclic groups) of odd prime-power order due to Xu et al. [21]. We describe first a simple family of arc-transitive dihedrants. For a group G and g ∈ G we denote by g R the permutation of G acting as x g R = xg, x ∈ G, and write H R = {h R | h ∈ H } for a subgroup H ≤ G, in particular, G R is the right regular representation of G. Example 1.1 (Arc-transitive dihedrants D pn , .) Let p be an odd prime, and let the dihedral group D2 pn be given by the presentation n

D2 pn = a, b | a p = b2 = baba = 1. Denote Z∗pn the multiplicative group of invertible elements in the ring Z pn . It is wellknow that Z∗pn ∼ = Z pn−1 ( p−1) . Let ∈ Z∗pn of order m. We define the Cayley graph m−1 D pn , = Cay(D2 pn , S), where S = ba, ba , . . . , ba . We claim that the graph D pn , is arc-transitive. Consider the automorphism σ of D2 pn defined by a σ = a and bσ = b. The set S is an orbit of σ . This implies in turn that σ is an automorphism of D pn , , the group G = (D2 pn ) R , σ ≤ Aut(D pn , ), and

finally, G acts transitively on the arcs of D pn , . Denote K n , K n,n , and K n,n − n K 2 the complete graph with n vertices, the complete bipartite graph whose color classes have n vertices, and the graph obtained from K n,n by deleting a 1-factor. The complement of a graph is denoted by c . The lexicographical product 1 [2 ] of graphs 1 and 2 is the graph with vertex set V (1 ) × V (2 ), and (u 1 , u 2 ), (v1 , v2 ) are adjacent if and only if either u 1 is adjacent to v1 in 1 , or u 1 = v1 and u 2 is adjacent to v2 in 2 . Let D be a symmetrical block design. The incidence graph L(D) is the bipartite graph whose color classes are the set of points and the set of blocks, and a point is adjacent to block if these are incident in D. The non-incidence graph L (D) is defined correspondingly. Our main result is the following theorem.

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Theorem 1.2 Let be a connected, arc-transitive Cayley graph over D2 pn , p is an odd prime. Then ∼ = [K dc ], where d is a divisor of 2 p n , 1 ≤ d < 2 p n , and one of the following cases holds. (a) = K (2 pn )/d , (b) d = 2 p m , and = Cay(Z pn−m , K ), K ≤ Z∗pn−m , |K | divides p − 1, and |K | is even. (c) d = p m , and is one of the following graphs: (c1) K pn−m , pn−m − p n−m K 2 , (c2) L(P G(r, q)), L (P G(r, q)), where P G(r, q) is the r -dimensional projective space over the finite field G F(q), (c3) L(H11 ), L (H11 ), where H11 is the unique Hadamard design on 11 points, (c4) D pn , , where ∈ Z∗pn is of order m > 1, m divides p − 1. An outline of the paper is as follows. In Sect. 2 we introduce the concept of a saturated block system for vertex-transitive graphs. As a result, our initial problem is reduced to the class of those graphs which are neither complete, nor have a non-trivial saturated block system. The graphs in this class are described in Theorem 2.4. Our approach toward Theorem 2.4 relies on techniques from permutation group theory. In this context we explore a relation of arc-transitive Cayley digraphs with Schur rings, or for short S-rings. Namely, if = Cay(G, S) is arc-transitive, then S is a so called basic set of the S-ring over G induced by the group Aut(). We devote Sect. 3 to S-rings, in particular, give the definition of an S-ring, collect its properties relevant for us, and recall a result of Wielandt [19] on S-rings over dihedral groups. The proof of Theorem 2.4 is carried out in two sections: in Sect. 4 we prove a theorem on minimal block systems of the graphs in question (see Theorem 4.1), and complete the proof in Sect. 5. 2 Saturated Block Systems In this section we describe a special block system of vertex-transitive graphs. We start with notation and terminology. Let G be a group acting on a set . For a subset ⊆ , denote by G the elementwise stabilizer of in G, while by G {} the set-wise stabilizer of in G. For g ∈ G {} = {g | g ∈ G we write g for the permutation of induced by g, and let G {} {} }. If G acts transitively on and B is a block system of G, then G B denotes the kernel of G acting on B, i.e., G B = {g ∈ G | ∀B ∈ B : B g = B}. A block B of G is minimal if it does not contain any non-trivial, proper block of G, and a block system B is minimal if it is generated by a minimal block. For a vertex-transitive graph , denote by Block() the set of all block systems of the group Aut(). There are two trivial block systems in Block(): the partition of V () into singletons, and the partition that consists of only the whole set V (). These will be denoted by E V () and by UV () , respectively. The set Block() is partially ordered by the relation , where B1 B2 if any block of B1 is contained in some block of B2 . The poset (Block(A), ) is a lattice, i.e., closed under the operations ∧ and ∨, where B1 ∧ B2 is the partition whose classes are the intersection of B1 -blocks

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with B2 -blocks, and B1 ∨ B2 is the partition whose classes are the minimal subsets being union of B1 -blocks and B2 -blocks. Definition 2.1 Let be a vertex-transitive graph and B be a block system of Aut(). We say that B is saturated if for every edge {u, v} ∈ E() the blocks of B containing u and v are distinct, and any vertex in one of these blocks is adjacent to any vertex in the other block. The trivial block system E V () is clearly saturated, and in what follows we say that a saturated block system B is non-trivial if B = E V () . Every B ∈ Block() induces the quotient graph /B, which has vertex set B, and two blocks B1 and B2 in B are adjacent if and only if there is a vertex in B1 adjacent to a vertex in B2 . It is easily seen that if B is saturated with block size b, then ∼ = (/B)[K bc ]. Thus the automorphism group Aut() satisfies Aut() ∼ = Aut(/B) Sb . In the next proposition some basic properties of saturated block systems are proved. Proposition 2.2 Let be a vertex-transitive graph, and B be a non-trivial saturated block system of . (i) The block system B is unique. (ii) The quotient graph /B has no non-trivial saturated block system. (iii) If in addition = Cay(D2n , S), n ≥ 3, then /B is a Cayley graph over either a dihedral or a cyclic group. Proof (i) Toward a contradiction assume that B1 and B2 are two saturated block systems in Block() such that B1 = B2 and B1 , B2 = E V () . We claim that the block system B := B1 ∨ B2 is also saturated. Indeed, choose an edge {u, v} of . Now, let Bu the block in B that contains u, and Bv be the block in B that contains v. Pick an arbitrary v ∈ Bv . Since B = B1 ∨ B2 , there is a chain of blocks B1 , . . . , Br such that the following hold. • For all i ∈ {1, . . . , r }, Bi ∈ B1 or Bi ∈ B2 . • The vertex v ∈ B1 and the vertex v ∈ Br . • For all i ∈ {1, . . . , r − 1}, Bi ∩ Bi+1 = ∅. It is easy to see that u is adjacent to any vertex in B1 . Since B1 ∩ B2 = ∅, also, u is adjacent to any vertex in B2 . Continuing on this manner we see that u is adjacent to any vertex in Br , in particular, adjacent to v . This implies that any vertex in Bu is adjacent to any vertex in Bv . From this Bu = Bv , and so B is saturated. Obviously, B1 ≺ B (i.e., B1 B and B1 = B). Let B ∈ B, and let us consider the the set-wise stabilizer Aut(){B} of B in Aut(). Since Aut() ∼ = Aut(/B) Sb , the restriction of Aut(){B} to B is (Aut(){B} ) B = Sym(B), and hence it is primitive on B. This together with B1 ≺ B imply that B1 = E V () , a contradiction. (ii) Let B¯1 be a saturated block system of /B. Let B1 be the block system of such that B B1 , and that B1 factors to B¯1 by B. Since both B and B¯1 are saturated we get that B1 is saturated. By (i), B1 = B, hence B¯1 = E V (/B) . (iii) That any block system of a Cayley graph over a group G consists of the right cosets of some subgroup H ≤ G is [1, Theorem 1.5A]. Let B consists of the right

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cosets of the subgroup H ≤ D2n . Now, H is either a cyclic or dihedral subgroup, and we set H to denote the subgroup of H generated by its elements of largest order. That is, H = H if H is a cyclic subgroup, and it is an index two subgroup otherwise. We observe that (D2n ) R ∩ Aut()B = (H ) R . Let D¯ be the permutation group D¯ ≤ Sym(B) induced by the action of (D2n ) R on B. Then D¯ ∼ = D2n /H . = (D2n ) R /G B ∼ The group D¯ acts transitively on B. Moreover, it acts regularly if H is a cyclic subgroup, and in this case /B is a Cayley digraph over a dihedral group. If H is a dihedral subgroup, then D¯ has a regular cyclic subgroup, and so /B is a Cayley digraph over a cyclic group.

Our next proposition is about saturated block systems of Cayley graphs. Proposition 2.3 A Cayley graph Cay(G, S) has a non-trivial saturated block system if and only if S is a union of right cosets of a subgroup H, 1 < H ≤ G. Proof Put = Cay(G, S). Assume first that B is a non-trivial saturated block system of . Let H be the subgroup 1 < H ≤ G that B consists of the right cosets of H . We show that H s ⊆ S for all s ∈ S, and so S is the union of right cosets of H . Since B is saturated no edge is contained in H (regarded as a block), and so we find S ∩ H = ∅. Let s ∈ S, and consider the blocks H and H s. The edge {1, s} connects these blocks, and hence 1 is adjacent to any element in H s, implying H s ⊆ S. For the converse implication assume that S is a union of a non-trivial subgroup of G. Define the subgroup H ≤ G as follows, H = {x ∈ G | Sx = S}. Now, we have 1 < H ≤ G. Put A = Aut(), and denote by A1 the stabilizer of the identity element 1 in A. We claim that the subset H ⊆ V () is fixed by A1 . Notice that, for x ∈ V () the set of neighbors of x equals N x () = Sx. Thus for x ∈ H and g ∈ A1 , Sx g = N x g () = (N x ())g = (Sx)g = S g = S. That is, x g ∈ H, and H is fixed by A1 . Second, we claim that the right cosets of H form a block system for A. For g ∈ G we write g R for the right translation of G by g, i.e., x g R = xg for all x ∈ G. Let x ∈ H and g ∈ A1 . Then the image x = 1x R g = x g is in H, implying x R g(x −1 ) R ∈ A1 , and hence x R g = g x R for some g ∈ A1 . Therefore, H R A1 = A1 H R , so A1 H R is a subgroup of A. Since A1 ≤ A1 H R , the orbit of A1 H R containing 1 is a block for A. This orbit is H, and the claim follows. Denote B the block system consisting of the right cosets of H . It can be readily seen that B is saturated, and this completes the proof.

Let p be an odd prime, and let = Cay(D2 pn , S) be a connected and arc-transitive dihedrant. Suppose that has a non-trivial saturated block system B with block size

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b. Then /B remains connected, arc-transitive, and ∼ = (/B)[K bc ]. By part (ii) of Proposition 2.2, the quotient graph /B has no non-trivial saturated block system, and by part (iii) it is a Cayley graph over either a cyclic or a dihedral group. In the first case the block size b = 2 p m , and /B is of order p n−m , and so described by [21, Theorem 4.5] (see also [5, Theorem 1]). This way we obtain the complete graph K pn−m , and the graphs in class (b) of Theorem 1.2. To derive the remaining non-trivial class (c) it remains to prove the following theorem. Theorem 2.4 Let be a connected, arc-transitive Cayley graph over D2 pn , p is an odd prime, and ∼ = K 2 pn . If has no non-trivial saturated block system, then it is isomorphic to one of the following graphs: (a) K pn , pn − p n K 2 , (b) L(P G(r, q)), L (P G(r, q)), where P G(r, q) is the r -dimensional projective space over the finite field G F(q), (c) L(H11 ), L (H11 ), where H11 is the unique Hadamard design on 11 points. (d) D pn , , where ∈ Z ∗pn is of order m > 1, m divides p − 1. 3 Schur Rings Let G be a group written with identity element 1, and denoteZ(G) the group ring of G over the ring Z. The elements of Z(G) are formal sums x∈G ax x, ax ∈ Z, and Z(G) has addition

ax x +

x∈G

bx x =

x∈G

(ax + bx )x,

x∈G

and multiplication

x∈G

ax x

·

bx x

=

x∈G

(ax b y )(x y) =

x,y∈G

(ax bx −1 y )y.

x,y∈G

The group ring Z(G) is also a Z-module with scalar multiplication a

x∈G

ax x

=

(aax )x, a ∈ Z.

x∈G

For a subset S ⊆ G we write S for the Z(G)-element x∈G ax x defined by ax = 1 if x ∈ S, and ax = 0 otherwise. Following Wielandt [20] such element S we call simple quantity in Z(G). The transpose of η = x∈G a x x ∈ Z(G) is defined by η = x∈G ax x −1 . Definition 3.1 A Schur ring (for short S-ring) S over G of rank r is a subring of the group ring Z(G) such that there exist subsets S1 , . . . , Sr of G satisfying the following axioms:

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• S has a Z-module basis of simple quantities: S = S1 , . . . , Sr . • S1 = {1}, and ri=1 Si = G. • For every i ∈ {1, . . . , r } there exists j ∈ {1, . . . , r } such that Si = S j . The simple quantities S1 , . . . , Sr are called the basic quantities of S, the corresponding sets S1 , . . . , Sr the basic sets of S. As S is closed under multiplication, for all i, j ∈ {1, . . . , r }, Si · S j =

r

pikj Sk

k=1

holds with non-negative integers pikj . The numbers pikj are called the structure constants of S. Examples of S-rings arise from permutation groups X ≤ Sym(G) such that X contains the right regular representation G R of G. Let X 1 be the stabilizer of 1 in X , and S1 = {1}, S2 , . . . , Sr be the orbits of X 1 . It was proved by Schur [16] (see also [20, Theorem 24.1]) that the simple quantities S i generate an S-ring over G. This S-ring we shall call the S-ring over G induced by X (this is also called the transitivity module of G induced by X ). A detailed description of S-rings arising in this manner is presented in [20, Chapter IV], here we mention only one property, of which we make use in the sequel. Proposition 3.2 Let X be a permutation group X ≤ Sym(G) such that G R ≤ X , and S be the S-ring over G induced by X . Then for any subgroup H ≤ G, H ∈ S if and only if the right cosets of H form a block system of X . Next, we turn to S-rings over dihedral groups. The following notations will be kept for the rest of the paper: For n ≥ 3 denote by Cn the unique cyclic subgroup of D2n of order |Cn | = n. The group Z∗n acts on Cn as Aut(Cn ) by the rule x → x k for x ∈ Cn , k ∈ Z∗n . The corresponding orbits of Z∗n are the sets d = {x ∈ Cn | |x| = d}, where x is the subgroup of Cn generated by x, and d runs over the set of positive divisors of n. For R ⊆ Cn and k ∈ Z∗n we let R k = {x k | x ∈ R}, and denote by (Z∗n ){R} the set-wise stabilizer of R in Z∗n , i.e. (Z∗n ){R} = {i ∈ Z∗n | R i = R}. The next result is due to Wielandt (see [19, 3.5]). Theorem 3.3 Let S be an S-ring over D2n , and S be a basic set of S. Then for every k ∈ Z∗n there exists a basic set S of S such that S ∩ Cn = (S ∩ Cn )k , and |S| = |S |.

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Our interest in S-rings comes from the simple observation that for an arc-transitive Cayley digraph = Cay(G, S), the set S is a basic set of the S-ring over G induced by the group Aut(). The next lemma follows from this observation and Theorem 3.3. Lemma 3.4 Let Cay(D2n , S) be an arc-transitive dihedrant, and d be a positive divisor of n such that S ∩ d = ∅. Then S ∩ d is an orbit of (Z∗n ){S∩Cn } . Proof Put K = (Z∗n ){S∩Cn } . As S is a basic set of an S-ring S, we may apply Theorem 3.3. Both S ∩ Cn and d are fixed by K , so S ∩ d is fixed by K too. Pick x, y ∈ S ∩ d , x = y. Since d is an orbit of Z∗n , there is k ∈ Z∗n such that x k = y. Now, let S be the basic set of S such that S ∩ Cn = (S ∩ Cn )k (see Theorem 3.3). Then y ∈ S ∩ S , S = S, and so k ∈ K . Hence K acts transitively on S ∩ d , and the lemma follows.

4 Minimal Blocks Systems Let = Cay(D2n , S) and B be a block system of with block size b. Then B consists of the right cosets of a subgroup H ≤ D2n and b = |H |. This shows that if b is odd, then B is uniquely defined by its size b. In what follows such block system we denote also by Bb . Our goal in this section is to prove the following theorem. Theorem 4.1 Let p be an odd prime, = Cay(D2 pn , S) be a connected, arc-transitive graph without a non-trivial saturated block system, and B be a minimal block system of with block size b, 2 < b < p n . Then B = B p and the kernel Aut()B ∼ = Zp. We give first four preparatory lemmas. Throughout the section p stands for an odd prime number. Let G ≤ Sym() be a transitive permutation group such that H ≤ G is a regular subgroup which is either cyclic or dihedral. As in the previous section, we denote by H the subgroup generated by the elements of H of largest order. Thus if H is dihedral of order greater than 4, then H is the unique cyclic subgroup of order |H |/2, and H = H if H is cyclic. The following result is [6, Lemma 2.1]. Lemma 4.2 Let G ≤ Sym() be a 2-transitive group, and H ≤ G be a regular subgroup, H is cyclic or dihedral, |H | > 4 and |H | is non-prime. Then each normal subgroup N ≤ G which contains H is 2-transitive. Lemma 4.3 Let = Cay(D2 pn , S) be a connected, arc-transitive graph without a non-trivial saturated block system. If has a block system with block size 2, then B pn ∈ Block(). Proof Let B be a block system of with block size 2. Then B consists of the con sets {1, b}x, x ∈ D2 pn , where D2 pn = a, b | a p = b2 = baba = 1. Suppose that g x1 , x2 ∈ S so that {1, b} x1 = {1, b} x2 . Then x2 = x1 for some g ∈ Aut()1 . As B ∈ Block(), g maps the set S ∩ {1, b} x1 to S ∩ {1, b} x2 , implying the existence of a constant c such that |S ∩ {1, b} x| = c, whenever S ∩ {1, b} x = ∅.

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If c = 2 then B is saturated, see Proposition 2.3, hence c = 1. We set the notations: S0 = Cn ∩ S, and S1 = {x ∈ Cn | xb ∈ S}. For s ∈ S denote by (s) the subgraph of induced by the vertices {1, b, s, bs}. This is a bipartite graph, with bipartition sets {1, b} and {s, bs}, and has the edge {1, s}. Assume first that s ∈ Cn . Using that s −1 ∈ S and that c = 1 we see that (s) has two edges {1, s} and {b, bs}. Second, let s ∈ D2n \ Cn , and write s = a i b. It follows in a similar way that (s) has two edges if a −i b ∈ S, and one edge otherwise. Arc-transitivity of yields that (s1 ) ∼ = (s2 ) for all s1 , s2 ∈ S, thus we conclude that S0 = ∅ implies that S1 = S1−1 . Let S0 = ∅. Then it follows easily that is a bipartite graph with color classes Cn and D2n \ Cn . This gives us B pn ∈ Block(), as required. Let S1 = S1−1 . Now, consider the involution t in Sym(D2n ) defined as t = (1, b)(a, ba)(a 2 , ba 2 ) · · · (a n−1 , ba n−1 ). It can be checked directly that t is in Aut(), and that t centralizes a R . Therefore, is a Cayley graph over the cyclic group t, a R ∼ = Z2 pn . In this case the lemma follows from [5, Theorem 1].

Lemma 4.4 Let = Cay(D2 pn , S) be a connected, arc-transitive graph without a non-trivial saturated block system, and B be a minimal block system of Aut() with block size b. Then one of the following cases holds: • Aut()B acts 2-transitively on every block B ∈ B. • b = 2 or b = p, and (Aut()B ) B ∼ = L ≤ AG L(1, b) for every block B ∈ B. Proof Put A = Aut() and N = Aut()B . For b = 2 or b = p the lemma follows by Burnside Theorem (see [1, Theorem 3.5B]). B is Let b > p, and B be a block in B. As B is minimal, the permutation group A{B} B is a regular cyclic subgroup. primitive. If the block size b = p m , then (C pm ) BR ≤ A{B} B B m If the block size b = 2 p , then D R ≤ A{B} is a regular dihedral subgroup, where D = (C pm ) R , x R and x ∈ B \ C pn . In what follows, we denote by H the respective B . Thus the permutation group A B is 2-transitive. In above regular subgroup of A{B} {B} fact, this was proved by Schur (see [1, Theorem 3.5A]) if H is a cyclic group, and by B . It Wielandt [19, Satz 2] if H is a dihedral group. Then N A{B} , hence N B A{B}

is easily seen that H ≤ N B , hence by Lemma 4.2, N B is 2-transitive. The next lemma is proved exactly on the same way as [11, Lemma 2.4]. Lemma 4.5 Let = Cay(D2 pn , S) be a connected, arc-transitive graph without a non-trivial saturated block system, and B be a minimal block system of Aut(). Then Aut()B acts faithfully on every block B ∈ B. Proof Toward a contradiction assume that N := Aut()B acts unfaithfully on a block B in B, and denote by K the kernel of N acting on B. Let x ∈ V () not fixed by

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K . Since is connected we have a path path x = x1 , . . . , xr such that xr ∈ B. Correspondingly, there is a sequence of distinct blocks B1 , . . . , Br such that xi ∈ Bi for all i. Since K B1 is non-trivial and K Br = K B is trivial, there can be chosen consecutive blocks Bi , Bi+1 such that K Bi is non-trivial while K Bi+1 is trivial. Since N Bi is primitive (see Lemma 4.4) and K Bi N Bi , K Bi is transitive on Bi . This gives that xi+1 is adjacent not only to xi , but to all the vertices in Bi . By arc-transitivity there is an automorphism switching Bi with Bi+1 , consequently, any vertex in Bi is adjacent to any vertex in Bi+1 . We conclude that B is a non-trivial saturated block system, a contradiction.

We are ready to prove the main result of the section. Proof of Theorem 4.1 It is enough to show that N := Aut()B ∼ = Z p . By Lemma 4.5, N acts faithfully on every block of B. We claim that N can have at most two inequivalent actions on the blocks of B. If N B is 2-transitive, then this is a consequence of the classification of 2-transitive permutation groups (see e.g. the exposition in [1, Section 7.7]). If N B is not 2-transitive, then N ∼ = NB ∼ = L < AG L(1, p), by Lemma 4.4, hence acts equivalently on all blocks of B. As b < p n , there are at least three blocks in B. We may choose blocks B1 , B2 ∈ B on which N acts equivalently, B1 ∩ C pn = ∅, and B2 ∩ C pn = ∅. Let x1 ∈ B1 such that x1 ∈ C pn . Then there exists x2 ∈ B2 such that N x1 = N x2 . Define the set F = Fix(N x1 ) = {x ∈ D2 pn | ∀g ∈ N x1 : x g = x}. Since N Aut(), it is not hard to prove that F is a block of (see also [6, Proposition 5.2]). Denote by F the block system of generated by F. Notice that, x1 , x2 ∈ F, so F = E V () . Observe that, if N ∼ = Z p , then N x1 has only one fix point in B1 , which is x1 . Then F ∩ B1 = {x1 }, implying B ∧ F = E V () . Using that b > 2 we obtain that B p B. This follows from the fact that B is given by the right cosets of a subgroup K ≤ D2 pn of order |K | = b, and that any subgroup of D2 pn of order larger than 2 contains the unique cyclic subgroup of order p. If F has block size larger than 2, then by the same reason B p F also holds, and hence B p B ∧ F. This contradicts that B ∧ F = E V () , and hence F has block size 2. This in turn implies that, / C pn , and B B pn . By Lemma 4.3, B pn ∈ Block(). Then we F = {x1 , x2 }, x2 ∈ conclude that B ∧ B pn ∈ Block(), B p B ∧ B pn , and B ∧ B pn ≺ B. A contradiction to the minimality of B. Therefore, N ∼

= Z p , and the theorem is proved. 5 Proof of Theorem 2.4 Throughout the section we use p for an odd prime number. As the first step to prove Theorem 2.4, we derive the graphs D pn , under the additional assumption that (C pn ) R is normal in Aut(). Lemma 5.1 Let = Cay(D2 pn , S) be a connected, arc-transitive graph, which has no non-trivial saturated block system. If (C pn ) R Aut(), then is isomorphic to D pn , , where ∈ Z ∗pn is of order m > 1, m divides p − 1.

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Proof As (C pn ) R Aut(), the graph is bipartite with color classes V1 = C pn and V2 = D2 pn \ C pn . Then B pn ∈ Block(), and we have N := Aut()B pn = Aut(){V1 } = Aut(){V2 } . We claim that N acts faithfully on both V1 and V2 . To the contrary assume that N acts unfaithfully on Vi , where i ∈ {1, 2}, with non-trivial kernel K . Then K N , hence K V j N V j , where j ∈ {1, 2} \ {i}, and K V j = 1. Thus any orbit B of K V j is a block of N V j with block size |B| > 1, and thus also a block of Aut(). It is not hard to prove that the block system B of Aut() generated by B is saturated, a contradiction. Since N acts faithfully on both V1 and V2 and (C pn ) R N , we conclude that N acts on both on V1 and V2 equivalently to its action on C pn by conjugation. Thus Aut()1 = N1 = Nb for some b ∈ D2 pn \C pn . From this we see that any g ∈ Aut()1 acts on D2 pn \ C pn as (ba i )g = ba ik for some k ∈ Z∗pn . This shows that the action of Aut()1 is described by a subgroup K ≤ Z∗pn , K = 1, and S = {ba i | a i ∈ }, where is an orbit of K such that = C pn . As Z∗pn is a cyclic group, K = for some ∈ Z∗pn . Using that has no non-trivial saturated block, we obtain easily that

|K | divides p − 1, and finally that ∼ = D pn , . The above proof gives us the following corollary. Corollary 5.2 Let = Cay(D2 pn , S) be a connected, arc-transitive graph, which has no non-trivial saturated block system. If B pn ∈ Block(), then Aut() acts faithfully on both blocks of B pn . A crucial observation in proving Theorem 2.4 is that B pn ∈ Block() holds for any graph in question. This we are going to prove in Lemma 5.4, but before give an auxiliary lemma. Lemma 5.3 Let n ≥ 2, and = Cay(D2 pn , S) be a connected, arc-transitive graph, which has no non-trivial saturated block system. Suppose, in addition, that ∼ = K 2 pn and B pn ∈ / Block(), and let C p denote the subgroup of C pn of order p. Then (i) |S ∩ C p x| ≤ 1 for all x ∈ D2 pn , and (ii) |S ∩ C pn | is a divisor of p − 1. Proof We set S0 = S ∩ C pn . (i) Let B be a minimal block of . If B has block size 2, then B pn ∈ Block(), which is not the case. By Theorem 4.1, B = B p , and the corresponding kernel is Aut()B ∼ = Zp. g Suppose that x1 , x2 ∈ S so that C p x1 = C p x2 . Then x2 = x1 for some g ∈ Aut()1 . As B p ∈ Block(), g maps the set S ∩ C p x1 to S ∩ C p x2 , implying the existence of a constant c such that |S ∩ C p x| = c, wheneverS ∩ C p x = ∅. We claim that c = 1. For otherwise, we can find x1 , x2 ∈ S ∩ C p x with x ∈ C pn . Then x1 , x2 ∈ pi for some i ∈ {2, . . . , n}. There exists k ∈ Z∗pn , x1k = x2 , and the

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orbits of k in pi are the right cosets of C p . Then S0k ∩ S0 = ∅, hence S0k = S0 , see Theorem 3.3, and k is in the the set-wise stabilizer (Z∗pn ){S0 } . But, S ∩ pi = S0 ∩ pi is an orbit of (Z∗pn ){S0 } , see Lemma 3.4. These imply in turn that C p x ⊂ S0 , c = p, and that B is saturated. A contradiction, hence c = 1, and (i) follows. (ii) Suppose that S0 ∩ pi = ∅ and S0 ∩ p j = ∅ for some i, j ∈ {2, . . . , n}, i < j. Then exists an k ∈ Z∗pn such that k fixes point-wise pi , and k has orbits the right cosets of C p acting on p j . Thus S0k ∩ S0 = ∅, and hence S0k = S0 , k ∈ (Z∗pn ){S0 } , and c = p, a contradiction. We conclude that S0 equals one orbit of the subgroup (Z∗pn ){S0 } . Using also that c = 1 this orbit must have size a divisor of p − 1, and (ii) follows.

Lemma 5.4 Let = Cay(D2 pn , S) be a connected, arc-transitive dihedrant such that ∼ = K 2 pn and does not have a non-trivial saturated block system. Then B pn ∈ Block(). Proof We prove the lemma by way of contradiction. For the rest of the proof let be a counter example with minimal exponent n. That is, = Cay(D2 pn , S) ∼ = K 2 pn , is connected, arc-transitive, has no non-trivial saturated block system, and B pn ∈ Block(); and also the lemma holds true for any graph with order 2 p n < 2 p n . Let n = 1. Then has a minimal block system with block size either 2 or p. In either case B p ∈ Block(), see Lemma 4.3, hence n ≥ 2 follows. Let B ∈ Block() be minimal. If B has block size either b = 2, or b = p n , then B pn ∈ Block(), therefore 2 < b < p n . Theorem 4.1 can be applied, we find B = B p and the kernel N := Aut()B p = (C p ) R . Consider the quotient /B p of induced by B p . Observe that, if /B p satisfies all assumptions in Lemma 5.3, then the minimality of yields that B pn−1 ∈ Block(/B p ). But, then it follows readily that B pn ∈ Block() too, a contradiction. Therefore, it remains to show that all assumptions in Lemma 5.3 apply to /B p . • /B p is a connected, arc-transitive Cayley graph over D2 pn−1 . The first two properties are straightforward, for the third see (iii) of Proposition 2.2. • /B p ∼ = K 2 pn−1 . To the contrary assume that /B p ∼ = K 2 pn−1 . By (i) of Lemma 5.3, |S ∩ C p x| = 1 for all x ∈ D2 pn \ C p . Thus |S0 | = |S ∩ C pn | = p n−1 − 1. Compare this with (ii) of Lemma 5.3 to find n = 2. Furthermore, S0 is an orbit of the subgroup of Z∗p2 of order p − 1. We shall denote this subgroup by K . Consider the S-ring S over D2 p2 induced by Aut(). Recall that S is a basic set of S. It can be confirmed quickly that the remaining basic sets of S are described as S1 = {1}, S2 , . . . , Sr , Sr +1 = S, Sr +2 , . . . , Sr + p , where S2 , . . . , Sr are contained in C p , these are the orbits of a subgroup of Z∗p2 . The further basic sets Si , i > r, all satisfy |Si ∩ C p x| = 1

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for all x ∈ (D2 p2 \ C p ),

(1)


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and the intersections Si ∩ C p2 are the orbits of the subgroup K ≤ Z∗p2 . Note that |Si | = u for all i ∈ {2, . . . , r }, u(r − 1) = p − 1. Now, assume that u = 1. Let C p2 = a. Then we may write S2 = {a p }, and Si , S j for the basic sets so that i, j ∈ {r + 1, . . . , r + p}, Si ∩ C p2 equals the orbit Si ∩ C p2 = a K , and S j ∩ C p2 = (a p+1 ) K . Then in S we calculate S2 Si = S j . Translating this into Z p2 we find K + p = ( p + 1)K (here addition and multiplication are in Z p2 ). As each of these sets contains exactly one element from a coset of the subgroup of Z p2 of order p, it follows x + p = x p + x for each x ∈ K . This is clearly impossible, hence u > 1. Next, write Sr +1 · Sr +2 = x∈D2 p2 cx x. Let us calculate the number γ = −1 = Si if i ∈ {r + 1, . . . , r + p} we x∈C p \{e} cx . From (1) and the fact that Si find γ = #{ (x, y) ∈ Sr +1 × Sr +2 | x y ∈ C p \ {e} } |Sr +1 ∩ C p y −1 | − |Sr +1 ∩ Sr−1 = +2 | = |Sr +2 | = 2 p − 1. y∈Sr +2

r + p On the other hand Sr +1 · Sr +2 = i=1 pri +1,r +2 Si , with structure constants pri +1,r +2 . Thus γ = ri=2 pri +1,r +2 |Si |, implying that u divides γ = 2 p − 1. But, u divides also p − 1, a contradiction to u > 1. • /B p has no non-trivial saturated block system. To the contrary let C be a non-trivial saturated block system of /B p . Then /B p ∼ = Cay(D2 pn /C p , S/C p ), where S/C p = {C p x | x ∈ S}. By Lemma 5.1, |S/C p ∩ C pn /C p | = |S ∩ C pn | is a divisor of p − 1, hence C has block size 2. This implies that n has a block system B of size 2 p. Let D2 pn = a, b | a p = b2 = baba = 1, and n−1 D = a p , b be its subgroup such that B = {D x | x ∈ D2 pn }. Put A = Aut(). Recall that N = (C p ) R is the kernel of A acting on B p , hence N A. The stabilizer A1 fixes the set D \ C p . Let us consider the permutation group X = (A1 )(D\C p ) ≤ Sym(D \ C p ). The group X normalizes N (D\C p ) ∼ = Zp. If X is not transitive, then it fixes a point, say d. Then {d} is a basic set of the S-ring S over D2 pn induced by A. Therefore, {1, d} ∈ S, hence {1, d} is a block of by Proposition 3.2, so B pn ∈ Block(), a contradiction. It remains to consider the case that X is transitive. Then A1 contains an element g of order p such that g (D\C p ) is a p-cycle. The quotient graph /B is a connected, arc-transitive circulant graph of order p n−1 , and of degree |S0 |. By (ii) of Lemma 5.3, |S0 | divides p − 1, and this in turn implies that g ∈ AB , and g fixes point-wise C p S. Take an involution s ∈ S \ C pn . Then the conjugate g s fixes point-wise D, while acts as a p-cycle on the coset C p bs. Therefore, g s ∈ A1 , and |S ∩ C p bs| = p, a contradiction.

Our last lemma, before we start proving Theorem 2.4, is a direct consequence of [2, Theorem 6.3].

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Lemma 5.5 Let X be a permutation group X ≤ Sym(C pn ), p is an odd prime, such that X is not 2-transitive, and (C pn ) R ≤ X is a Sylow p-subgroup of X . Then (C pn ) R X . Proof of Theorem 2.4 Recall that = Cay(D2 pn , S) is a connected, arc-transitive graph, ∼ = K 2 pn , and has no non-trivial saturated block system. By Lemma 5.4, B pn ∈ Block(). Put A = Aut() and N = AB . The group N acts faithfully on both blocks in B pn , see Corollary 5.2. CASE 1. N acts 2-transitively on C pn . If N acts equivalently on these blocks, then the stabilizer A1 = N1 fixes a point in D2 pn \ C pn , say b, and is transitive on the rest (D2 pn \ C pe ) \ {b}. Therefore, S = (D2 pn \ C pe ) \ {b}, and it is easily seen that ∼ = K pn , pn − p n K 2 as given in (a) of Theorem 2.4. Let N act inequivalently on the blocks of B pn . It is proved in [11, Remark 2.6] that in this case is isomorphic to the incidence/non-incidence graph of a symmetric block design D. Now, D has a 2-transitive automorphism group having a regular cyclic subgroup, and being so, D is either the r -dimensional projective space P G(r, q) over the finite field G F(q), or the unique Hadamard design H11 on 11 points (see [4]). The respective cases are (b) and (c) of Theorem 2.4. CASE 2. N does not act 2-transitively on C pn . We claim that in this case (C np ) R A = Aut(). Then Lemma 5.1 applies to , and we get (d) of Theorem 2.4. If n = 1 then the claim follows by Burnside Theorem. Let n ≥ 2. By Theorem 4.1, the minimal block system B B pn is B = B p , and (C p ) R A. Let P denote a Sylow p-subgroup of A such that (C pn ) R ≤ P. Assume that (C pn ) R = P. Then N P ((C pn ) R ) > (C pn ) R . Therefore we can choose g ∈ N P ((C pn ) R ) which fixes 1 and is of order p. It is proved in [6, Proposition 6.2] that g ∈ N A ((D2 pn ) R ), hence g is in Aut(D2 pn ). Thus g fixes all cosets C p x set-wise. Suppose that g fixes an n−1 involution b ∈ D2 pn . Then g fixes all elements in the subgroup D = a p , b, and acts as a p-cycle on every coset C p x ⊆ D. Since is connected, |S ∩ C p x| = p for some x ∈ D2 pn \ D. But as B p ∈ Block(), we find |S ∩ C p x| = p whenever S ∩ C p x = ∅, a contradiction. Therefore P = (C pn ) R . The group N acts faithfully on C

n

C pn , N C pn is not 2-transitive, and it has a regular cyclic Sylow p-subgroup, (C pn ) R p . C

n

By Lemma 5.5, (C pn ) R p N C pn , (C pn ) R N . As (C pn ) R is characteristic in N , and N A, we find (C pn ) R A, as claimed. This completes the proof of the theorem.

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