1 Introductory remarks Throughout this paper graphs are nite, simple and undirected. Adopting the terminology of Tutte [11], a k-arc in a graph X is a

Transcription

1 ON 2-ARC-TRANSITIVE CAYLEY GRAPHS OF DIHEDRAL GROUPS Dragan Marusic 1 IMFM, Oddelek za matematiko Univerza v Ljubljani Jadranska 19, 1111 Ljubljana Slovenija Abstract A partial extension of the results in [1], where 2-arc-transitive circulants are classied, is given. It is proved that a 2-arc-transitive Cayley graph of a dihedral group is either a complete graph or a bipartite graph. Keywords: permutation group, imprimitive group, dihedral group, Cayley graph, 2-arc-transitive graph. 1 Supported in part by the Ministrstvo za znanost in tehnologijo Slovenije, project no. J

2 1 Introductory remarks Throughout this paper graphs are nite, simple and undirected. Adopting the terminology of Tutte [11], a k-arc in a graph X is a sequence of k + 1 vertices v 1 ; v 2 ; : : : ; v k+1 of X, not necessarily all distinct, such that any two consecutive terms are adjacent and any three consecutive terms are distinct. A graph X is said to be k-arc-transitive if the automorphism group of X, denoted Aut X, acts transitively on the k-arcs of X. Let X be a 2-arc-transitive graph and let v 2 V (X). It is not dicult to see that the restriction of the vertex stabilizer (Aut X) v to the set of neighbors N(v) of v is a doubly transitive group (see [9, Lemma 9.4]. As a byproduct of the classication of nite simple groups all doubly transitive groups are known [3]. With this information an organized eort has been set in motion to understand the structure of 2-arc-transitive graphs via an analysis of possible (doubly transitive) restrictions of the corresponding vertex stabilizers, resulting in a series of papers dealing with particular families of doubly transitive groups [2, 5, 6] (see [9] for a complete survey). From a somewhat dierent perspective, in [1] a classication of 2-arc-transitive circulants, that is Cayley graphs of nite cyclic groups, was given. Proposition 1.1 ([1, Theorem 1.1]) A 2-arc-transitive circulant is either a cycle or a complete graph, or a complete bipartite graph, or the complete bipartite graph minus a 1-factor K m;m? mk 2, where m is odd. Here we generalize the above result by partially extending it to 2-arctransitive Cayley graphs of dihedral groups. The following is thus the main result of this paper. Theorem 1.2 A connected, 2-arc-transitive Cayley graph of a dihedral group of order 2n, n 3, is either (i) the cycle C 2n, which is k-arc-transitive for any k 2; or (ii) the complete graph K 2n, which is 2-arc-transitive but not 3-arc-transitive; or (ii) a bipartite graph of valency at least 3. In Section 2 we describe some of the tools and conventions and prove some results that are needed in the proof of Theorem 1.2, carried out in Section 3. 2

3 2 Preliminary concepts Letting S be an arbitrary subset of Z n nf0g such that S =?S, and letting T be an arbitrary subset of Z n, we denote by X(2n; S; T ) the graph with vertex set fu i : i 2 Z n g [ fv i : i 2 Z n g, and edges of the form u i u i+s, v i v i+s, for all i 2 Z n and s 2 S, and u i v i+t, for all i 2 Z n and t 2 T. The ordered pair (S; T ) is said to be a symbol of X(2n; S; T ). Note that Aut X(2n; S; T ) contains a regular dihedral group generated by the automorphisms and mapping according to the rules: and u i = u i+1 ; v i = v i+1 ; i 2 Z n ; u i = v?i ; v i = u?i ; i 2 Z n : On the other hand, it is not dicult to see that for every Cayley graph X of a dihedral group with 2n elements there exists an ordered pair (S; T ), where S is a symmetric subset of Z n nf0g and T is a subset of Z n, such that X is isomorphic to the graph X(2n; S; T ). For the purpose of this paper a Cayley graph of a dihedral group will be referred to as a dihedrant and any corresponding ordered pair (S; T ) in the above denition as its symbol. Proposition 2.1 Let X = X(2n; S; T ) be a connected dihedrant of degree at least 3. Then the girth of X is (i) at most 4 if S 6= ;; (ii) and at most 6 if S = ;. Proof. Note that T 6= ; since X is connected. To prove (i), let s 2 S and t 2 T. Then u 0 ; u s ; v s+t ; v t ; u 0 is a 4-cycle in X. Next, to prove (ii), let t; t 0 ; t 00 2 T be distinct. Then u 0 ; v t ; u t?t 0; v t?t0 +t 00; u t 00?t 0; v t 00; u 0 is a 6-cycle of X. The next result is taken from [1, Proposition 1.2] and allows us to consider only dihedrants of girth 4 or more. Proposition 2.2 If X is a connected 2-arc-transitive graph of girth 3, then X = K n for some n. 3

4 Let Y be a graph, let k be a positive integer, and let f : A(Y )?! S k be a permutation voltage assignment, that is a function from the set of arcs of Y into the symmetric group S k where reverse arcs carry inverse voltages. We thus have a labeling of the arcs of Y by permutations in S k such that f u;v f v;u = id for all pairs of adjacent vertices u; v in Y, where f u;v denotes the permutation assigned to the arc (u,v). The voltage assignment f naturally extends to walks in Y. In particular, for any cycle C = v 1 v 2 : : : v r v 1 of Y we let f C denote the voltage f v1 ;v 2 f v2 ;v 3 : : : f vr;v 1 of C, that is the f-voltage of C. The covering graph X = Cov(Y; f) of Y with respect to f has vertex set V (Y ) Z k, and edges of the form (u; r)(v; s), where u; v 2 E(Y ), r 2 Z k and s = r f u;v. The graph Y is said to be the base graph of X, and the latter is sometimes referred to as a k-fold cover of Y. The set of vertices (u; 0); (u; 1); : : : ; (u; k? 1) is called the bre of u. The subgroup of all those automorphisms of X which x each of the bres setwise is called the group of covering transformations. The following result needs no proof. Proposition 2.3 The group of covering transformations of a connected covering graph acts semiregularly on each of the bres. In particular, if the group of covering transformations is regular on the bres of X = Cov(Y; f), we say that X is a regular cover. In this case, the voltage group Im(f) is a regular group of degree k abstractly isomorphic to the group of covering transformations. The next result is immediate. Proposition 2.4 If X = Cov(Y; f) is a 2-fold cover then it is a regular cover. The proposition below gives a standard result in the theory of covering spaces. We omit its proof. Proposition 2.5 Let Y be a graph, let X = Cov(Y; f) be a regular covering of Y with K as the group of covering transformations, and let 2 Aut Y be such that its lift ~ centralizes K. Then f C = f C any cycle C of Y. The canonical double cover CDC(Y ) of a graph Y is the 2-fold cover Cov(Y; f) where f assigns to each arc in Y the nonidentity element of S 2. The following result will be needed in the proof of Theorem

5 Lemma 2.6 Let m 4 be a positive integer, and let X be a 2-fold cover of Y = K m with girth greater than 3. Then X = CDC(K m ) = K m;m? mk 2. Proof. Firstly, by Proposition 2.4, we have that X is a regular cover with respect to the group Z 2. Without loss of generality we may assume that the spanning tree of Y all of whose edges are assigned voltages 0 is a star K 1;m?1. But the girth of X is at least 4 and so the voltages of all remaining edges of Y equal the generator of the group of covering transformations isomorphic to Z 2. By a relabeling of the vertices belonging to the bre corresponding to the center of the star we easily see that X = CDC(K m ) = K m;m? mk 2. Lemma 2.7 Let Y be a graph with a permutation voltage assignment f. If T is a triangle of Y contained in some K 4 all of whose 4-cycles have identity voltage, then (f T ) 2 = id. Proof. Let T = a; b; c; a and let = f a;b ; = f b;c, and = f c;a. Let x be the fourth vertex of a K 4 satisfying the assumptions of this lemma. Let = f a;x ; = f b;x, and = f c;x. Then the fact that the voltage of each 4-cycle in Y is id implies?1 =?1 =?1 = id. Hence (f T ) 2 = () 2 = id. 3 Proving Theorem 1.2 The following group-theoretic results will play a crucial role in the proof of Theorem 1.2. Proposition 3.1 (Burnside, [12, Theorems 25.3 and 25.6]) A primitive group containing a regular cyclic subgroup of composite order or a regular dihedral subgroup is doubly transitive. Proposition 3.2 ([10, ]) Let H be a subgroup of a group G. Then C G (H) is a normal subgroup of N G (H) and the quotient N G (H)=C G (H) is isomorphic with a subgroup of Aut H. The following result may be deduced from the classication of doubly transitive groups (see [3] or [4, p.243]). 5

6 Proposition 3.3 A 3-transitive but not 4-transitive group may be identied with one of the following groups. (i) M 11 of degree 12; (ii) M 22 of degree 22; (iii) P SL(2; q) G P?L(2; q), where P SL(2; q) is the socle of G, and G acts on P G(1; q) in a natural way, having degree q + 1, with q a prime power. Finally, we have Lemma 3.4 Let G be a nite group and let K be a normal subgroup of G which is either cyclic or isomorphic to Z 2 Z 2 nonabelian simple group. Then C G (K) = G. and such that G=K is a Proof. Clearly, C G (K) K, for K is abelian. Moreover, C G (K) is normal in G and hence C G (K)=K is normal in G=K. But G=K is simple and therefore C G (K)=K = 1 or C G (K)=K = G=K. In the rst case C G (K) = K and so, by Proposition 3.2, we have that G=K = G=C G (K) is isomorphic to a subgroup of Aut K. But this is not possible in view of our assumptions on K. Therefore C G (K) = G. We adopt the following notation. For a permutation group G acting on a set V and a subset W of V we let G W denote the setwise stabilizer of W in G and we let G fw g denote the pointwise stabilizer of W in G. Also, we shall say that a subset B of V is a minimal block for G if it is a block for G containing at least two points, and no other block with at least two points is properly contained in B. Proof of Theorem 1.2. The proof that the transitivities of the graphs C 2n and K 2n in Theorem 1.2 are as claimed, is left to the reader. Let X 6= C 2n ; K 2n, where n 3, be a connected 2-arc-transitive dihedrant. We will show that X is a bipartite graph. The proof is by induction on the order of X. Assume that the statement of the theorem has been proved for all dihedrants of order less than 2n. Let (S; T ) be a symbol of X. We may assume that S 6= ; and so, combining Propositions 2.1 and 2.2, it follows that 6

7 X has girth 4: (1) Next, since X 6= K 2n, Proposition 3.1 implies that A = Aut X is imprimitive. Choose a minimal block of A, say of cardinality k 2, and let B be the corresponding complete system of imprimitivity of A. (Clearly, A B B is primitive for each B 2 B.) Let K denote the kernel of the action of A on B and let A = A=K denote the corresponding quotient group. Moreover, let X denote the quotient graph X=B of X relative to B, that is the graph with vertex set B such that two blocks B and B 0 are adjacent if and only if there is an edge in X with one end-vertex in B and the other end-vertex in B 0. The 2-arc-transitivity of X then implies that X admits a 2-arc-transitive action of A: (2) Moreover, we have that each block in B induces a totally disconnected graph. Now, we may assume that jbj > 2, for otherwise the graph X is bipartite and we are done. Next, the fact that X is 2-arc-transitive, implies that each bipartite graph induced by two adjacent blocks in X, is isomorphic to kk 2. If that was not the case, then letting B 0 ; B 00 be neighbors of B in X we could nd 2-arcs in X of the form v 0 ; v; v 00 and v 0 ; v; u 0, with v 2 B; u 0 ; v 0 2 B 0 ; v 00 2 B 00, obviously not belonging to the same orbit of A. Therefore we have that X = Cov( X; f) is a k-fold cover of a connected 2-arc-transitive graph X. Since X 6= C 2n, the valency val X of X is at least 3, and of course X cannot be a cycle. Besides, if X is bipartite, then X must also be bipartite and we are done. We may therefore assume that X is neither a cycle nor a bipartite graph. (3) Moreover, if val X = 3 then, since S 6= ;, it is easily seen that X must be isomorphic to the cube Q 3 = K4;4? 4K 2, a bipartite graph. Hence we may assume that val X 4 and thus jbj 5: (4) Next, recall that since X is a connected graph, the action of K is semiregular on each block, in view of Proposition 2.3. Furthermore, since the orbits of K form a block system of G, the minimality of blocks in B then implies that K B is either regular or trivial for each B 2 B. In fact, we are going 7

8 to show that K is regular on each block in B; in other words, the group of covering transformations is regular on the bres B 2 B and X is thus a regular cover of X. More precisely, we will prove that two essentially dierent cases may occur with the restriction K B, B 2 B, isomorphic to either Z p, where p is a prime, or to Z 2 2. To see this let D denote a regular dihedral subgroup of A generated by an element of order n and an involution such that =?1. Of course, B is also a block system of D. Choose a block B 2 B and consider the setwise stabilizer D B. In view of (4), the latter is a subgroup of D of index at least 5. Therefore there exists m 2 Z n nf0g such that D B is either h m i or h m i or h m ; i. Suppose rst that D B = h m i. Then X must be a 2-fold cover and as such a regular cover of X by Proposition 2.4. Besides, it is easily seen that B = fb i : i 2 Z n g. Hence A contains a regular cyclic group hi and so X is a circulant, which is 2-arc-transitive by (2), and thus, combining together Proposition 1.1 and (3), we may assume that X is a complete graph, and so X = K n=2, forcing X to be bipartite by Lemma 2.6 and (1). Suppose next that D B = h m i. Then D B is a normal subgroup of D and so it clearly xes every block in B, forcing D B K. Of course, D B is regular on B and, since K is by Proposition 2.3 semiregular on each block, we have that K = D B is regular on each block in B. Besides, since K is a cyclic normal subgroup of A, every subgroup of K is normal in A, and thus its orbits form a block system of A. The minimality of blocks in B then forces k = p, a prime and K = Z p.. Thus X is a regular cover of X with respect to the cyclic group Z p. Moreover, we have that DK=K = D=K = D=D \ K is a regular dihedral subgroup of A and so X is itself a dihedrant, of order 2m, where m = n=p < n. Therefore by induction hypothesis we have that X, as a 2-arc-transitive dihedrant, satises the assumptions of Theorem 1.2. In view of (3) we thus have X = K 2m as the only possibility. Also, in this case we may assume that p is an odd prime in view of Lemma 2.6. Supose now that D B = h m ; i. (In this case D B is not a normal subgroup of D, unless n is even and m generates a subgroup of index 2 in hi.) It is easily seen that B = fb i : i 2 Z m g and thus the subgroup generated by m xes each of the blocks in B, that is id 6= h m i K. Hence K is nontrivial and thus it has to be regular on blocks in B. In particular, K B is a 2-extension of the cyclic group h m i B and thus the minimality of the block B forces jbj = 4, m = n=2 and so K = Z 2 2, for in all other cases the orbits of h m i would form an imprimitivity block system of A. Moreover, we have 8

9 that hi=k \ hi = hi=h m i is a regular cyclic subgroup of A and so X is a circulant, of order n=2. As in the rst case above, combining Proposition 1.1 and (3), we may thus assume that X is a complete graph. More precisely, X = K n=2. To summarize, we are now left with one of the following two situations. Either there exists an odd prime p such that K = Z p and the graph X is a regular p-fold cover of the complete graph K 2m, where m = n=p, with respect to the group Z p, or else K = Z 2 2 and X is a regular cover of K n=2 with respect to the group Z 2 2. Moreover, the quotient X admits a regular dihedral group in the rst case, and a regular cyclic group in the second case. In what follows we shall prove that X is necessarily bipartite in the case K = Z 2 2, and that, on the other hand, no connected cover satisfying the above assumptions exists in the case K = Z p. Fix a block B 2 B. By a relabeling of the vertices of X, if necessary, we may see that X = Cov( X; f), where f assigns voltage id to all the arcs having B as an end-vertex. In other words, we have chosen the spanning tree in X with id voltages to be the star with center in B. For the rest of the proof of Theorem 1.2, we shall now identify X with Cov( X; f). By a voltage of an arc (or a cycle) we shall mean the f-voltage of that arc (or cycle). Combining (2) with the fact that X is a complete graph, we have that A acts 3-transitively on the vertices of X, that is on the set B. If A is at least 4-transitive, then since, by (1), there must be at least one 4-cycle in X with voltage equal to id, it follows that all 4-cycles must have the same property. We can then use Lemma 2.7 to deduce that for all triangles T of X, the voltage f T is an involution. In particular, this implies that the voltage of each arc in X joining two blocks dierent from B is an involution, which immediately excludes the rst case K = Z p, p odd prime. Besides, letting B 0 ; B 00 ; B 000 be any three blocks dierent from B, we must have that f B = f 0 B ;B00 00 ;B 000, since the voltage of the 4-cycle B; B0 ; B 00 ; B 000 ; B is id. This forces all arcs with end-vertices diferent from B to have the same voltage. But then X = 2CDC(K n=2) would be disconnected, and so also the second case K = Z 2 2 leads to a contradiction. We conclude that A is precisely 3-transitive. Therefore A may be identied with one of the groups in Proposition 3.3. Observe that neither M 11 nor M 22 contains a regular subgroup of degrees 12 and 22, respectively, in the above 3-transitive actions. We may therefore identify the group A with a subgroup of P?L(2; q) having P SL(2; q) as a socle, and we may identify the set B with the projective line P G(1; q) = 9

10 GF (q) [ f1g. In particular, 2m = q + 1 and q = r a is a power of a prime r. Let G be the preimage of the subgroup G of A isomorphic with P SL(2; q). We will now analyze the action of G on X in more detail. Using Lemma 3.4 it follows that C G (K) = G when K = Z p as well as when K = Z 2 2. If q is even we have that G is 3-transitive on X, and so all triangles in X have the same voltage, which has to be an involution. But then this involution is the voltage of any arc in X, except for those of the spanning star with center in B (which have voltage id). In particular K = Z 2 2, and moreover X = 2CDC(K n=2 ) is disconnnected, a contradiction. Therefore q is odd and G is not 3-transitive on X. The analysis now depends on whether G contains an element reversing the orientation of a given triangle or not. (It may be seen that such an orientation reversing automorphism exists if and only if q 1 (mod 4) but this information is not really needed here.) If that is the case then, using the same argument as above, we have that the voltage of a triangle in X is an involution, forcing K = Z 2 2, and that, moreover, at most two dierent voltages for these triangles are possible. Hence all of the arcs in X not incident with B attain one of these two voltages This forces X to be bipartite, for every closed walk in X of odd length contains an even number of one and an odd number of the other of these two involutions. We may now assume that G contains no element reversing the orientation of a given triangle. But then G is transitive on 3-subsets in B, that is on unordered triangles in X. If K = Z 2 2 then again X = 2CDC(K n=2 ), a contradiction. Hence K = Z p and so, combining Proposition 2.5 together with (1), we have that there exists w 2 Z p such that for every triangle T of X f T 2 fw; p? wg: (5) Consider the action of the subgroup K G fbg = K GfBg of C G (K) on the subgraph, say Y, of X consisting of all the vertices in B and all the vertices in all the blocks belonging to an orbit of an element 2 G fbg, of order r in its action on B. Label these blocks as B i ; i 2 Z r, with B i+1 = B i. Let be a generator of K. Note that xes each vertex in B and has p orbits of length r in Y, that is precisely all of the neighbors' sets N(u) \ V (Y ), where u 2 B. Fix a vertex v 2 B and label it with the ordered pair (1; 0). Next, label its neighbor in B i, i 2 Z r, with (i; 0). Then let (1; s) = (1; 0) s 10

11 for all s 2 Z p and let (i; s) = (i; 0) s for all i 2 Z r and all s 2 Z p. The automorphisms and then map according to the rules: (s; i) = (s; i + 1) for all i 2 f1g [ Z r and s 2 Z p, and (s; i) = (s + 1; i) for all i 2 Z r and s 2 Z p, and xes each (1; i), i 2 Z r. Let w d = f B0 ;B d for all d 2 Z r. Since =, it follows that f Bi ;B i+d = w d for all i 2 Z r. With no loss of generality assume w 1 = w. If r = 3 then the triangle B 0 ; B 1 ; B 2 ; B 0 has voltage 3w and so by (5) we have 3w 2 fw; p? wg, forcing either 2w = 0 or 4w = 0. But both possibilities contradict the fact that p is an odd prime. Assume that r 5. Now the triangle B 0 ; B 1 ; B 2 ; B 0 has voltage 2w 1? w 2 = 2w? w 2 which must be either w or p? w. In the latter case w 2 = 3w 2 fw; p? wg. Thus either 4w = 0 or 2w = 0, again contradicting p odd. Therefore the rst case occurs and so w 2 = w 1 = w. Now, considering the triangle B 0 ; B 1 ; B 3 ; B 0, we obtain as above that its voltage 2w? w 3 must be equal to w rather than p? w, forcing w 3 = w. Continuing this way we have that w (p?1)=2 = w which forces the triangle B 0 ; B (p?1)=2 ; B (p+1)=2 ; B 0 to have voltage 3w. But then 3w = w or 3w = p? w, and thus again either 2w = 0 or 4w = 0, a contradiction. These contradictions show that X cannot be a p-fold cover of X if p is an odd prime, thus completing the proof of Theorem 1.2. Acknowledgement: The author is grateful to dr. Shao-fei Du for the helpful discussions during the preparation of this paper. References [1] B. Alspach, C. D. E. Conder, D. Marusic, M.-Y. Xu, A classication of 2-arctransitive circulants, J. Alg. Combin. 5 (1996), 83{86. [2] R. W. Baddeley, Two-arc transitive graphs and twisted wreath products, J. Alg. Combin. 2 (1993), 215{237. [3] P.J.Cameron, Finite permutation groups and nite simple groups, Bull. London. Math. Soc., 13 (1981), 1{22. [4] J. D. Dixon, B. Mortimer, \Permutation Groups", Springer-Verlag, New York,

12 [5] X. G. Fang, C. E. Praeger, Finite two-arc-tarnsitive graphs admitting a Suzuki simple group, Research Report, Department of Mathematics, University of Western Australia, (1996). [6] X. G. Fang, C. E. Praeger, Finite two-arc-tarnsitive graphs admitting a Ree simple group, Research Report, Department of Mathematics, University of Western Australia, (1996). [7] B. Huppert, \Endliche Gruppen I", Springer-Verlag, Berlin, [8] D.S. Passman, \Permutation Groups", Benjamin, Menlo Park, California, [9] C. E. Praeger, Finite transitive permutation groups and nite vertextransitive graphs, Symmetry in graphs, NATO Conference, Montreal, July 1997, to appear. [10] D. J. Robinson, \A course in the theory of groups", Springer Verlag, New York, [11] W. T. Tutte, On the symmetry of cubic graphs, (1958), [12] H. Wielandt, \Finite Permutation Groups", Academic Press, New York,

13 ON 2-ARC-TRANSITIVE CAYLEY GRAPHS OF DIHEDRAL GROUPS Keywords: permutation group, imprimitive group, dihedral group, Cayley graph, 2-arc-transitive graph. 13

14 ON 2-ARC-TRANSITIVE CAYLEY GRAPHS OF DIHEDRAL GROUPS Contact address: Dragan Marusic IMFM, Oddelek za matematiko Univerza v Ljubljani Jadranska 19, 1111 Ljubljana Slovenija tel fax dragan.marusic@uni-lj.si 14