A graphical characterization of the group Co 2


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1 A graphical characterization of the group Co 2 Hans Cuypers Department of Mathematics Eindhoven University of Technology Postbox 513, 5600 MB Eindhoven The Netherlands 18 September 1992 Abstract In this note we show that, up to isomorphism, there is a unique graph which is locally the distance 1 or 2 graph of the dual polar graph related to P SU 6 (2). The automorphism group of this graph is the second Conway group Co Introduction It is well known, see for example [3,6,7,8], that the second Conway group Co 2 is the flagtransitive automorphism group of a geometry with Buekenhout diagram c in which the residue of a point is the classical dual polar space related to P SU 6 (2). The point graph G of this geometry is a graph on 2300 points, such that for each vertex of this graph, the subgraph induced on the neighbors of the vertex, is the distance 1 or 2 graph of the dual polar graph related to P SU 6 (2). Denote by H this dual polar graph, and by H 1,2 its distance 1 or 2 graph. The group Co 2 acts as a rank 3 permutation group on the graph G, with point stabilizer P ΣU 6 (2). In this note we show that there is a unique graph which is locally H 1,2. Theorem 1.1 Let Γ be a connected graph, such that for each vertex v of Γ, the induced subgraph Γ v on the neighbors of v is isomorphic to H 1,2, then Γ is isomorphic to G. Meixner [6] and Yoshiara [8] have given characterizations of the geometry mentioned above, under the assumption that the automorphism group acts flagtransitively. 2. The dual polar graph related to P SU 6 (2) In this section we investigate the dual polar graph H related to the group P SU 6 (2). This graph is the point graph of a near hexagon with parameter set (s, t, t 2 ) = (2, 20, 4) and is distance regular with intersection array {42, 40, 32; 1, 5, 21}. We often identify the graph H with the near hexagon. 1
2 As we are interested in graphs which are locally the distance 1 or 2 graph of H, which we denote by H 1,2, we give for convenience of the reader the distribution diagram of this latter graph as seen from a vertex The geodesic closure of any two points at distance two in the near hexagon H is a generalized quadrangle of order (2,4) isomorphic to the dual of the quadrangle of singular points and lines in a nondegenerate 3dimensional unitary space. These generalized quadrangles are called quads. Every point outside a quad is collinear to exactly one point in the quad, and to every quad q we can attach an involution t q in the automorphism group of H, that fixes all points of the quad, and for each line meeting it in a point, interchanges the two points on the line that are not in the quad q. These involutions form a set of 3transpositions in Aut(H), the class of transvections in P SU 6 (2). A geometric hyperplane of the near hexagon H (or any other pointline geometry) is a set of points meeting every line in 1 or all points. A point of a geometric hyperplane is called deep, if all lines on the point are contained in the geometric hyperplane. In the following proposition we characterize a class of geometric hyperplanes of H, that appear as the complement of the µgraphs in the graph G. Proposition 2.1 Let H be a proper geometric hyperplane of H, in which every point is deep or on 15 lines in the hyperplane, then H is one of the orbits of length 567 on the points of H under a subgroup of Aut(H) isomorphic to O 6 (3). Proof. Let H be a hyperplane as in the hypothesis. Suppose q is a quad in H. Then H intersects q in a hyperplane of q, which is either q, a subquadrangle of order (2, 2) in q, or a point and all its neighbors in q. Let l be a line inside H, and suppose x and y are distinct points on this line. There are 5 quads on this line, the intersection of any two of them just being l. Since there are at least 15 lines through each of x and y in H, there are at most 3 quads on l that are not contained in H. Thus any line of H is contained in at least 2 quads inside H. Let q be a quad inside H. Then each point p of H is collinear with a unique point p of q, and the line through p and p lies in H. This implies that subgraph of H induced on H is connected, and that the involution t q, which maps p to the third point on the line through p and p, stabilizes H. 2
3 Now suppose p and p are two collinear points on the line l in H. Let m be a line through p, the third point on l, different from l. Then there is a quad q in H containing m, but not l. The involution t q maps p to p. In particular, as H is connected, the group G generated by the involutions t q, with q a quad inside H, is transitive on H. So H contains no deep points, and by counting pairs of collinear points (x, y) with x H and y H, we obtain that 12 H = 21(891 H ), so that H = 567. Moreover, G contains 567.x/27 = 21x involutions t q, where x is the number of quads in H on each point. As we saw before, this number x is between 15.2/5 = 6 and 15.3/5 = 9. Using the Atlas [5], or with the help of Fischer s classification of groups generated by 3transpositions, it is easy to see, that G is isomorphic to O 6 (3). This also follows by the results of [2]. This ends the proof of the proposition. The set of hyperplanes of H satisfying the hypothesis of the above proposition will be denoted by GH. The group P SU 6 (2) contains three conjugacy classes of subgroups isomorphic to O 6 (3) and generated by transvections. The outer automorphism group of P SU 6(2) induces an S 3 on these 3 classes. This implies that the group P SU 6 (2) has 3 orbits of length 1408 on the set GH of geometric hyperplanes satisfying the conditions of the previous proposition. For each H GH the set of lines in H induces a near hexagon on the 567 points of H, the Aschbacher near hexagon, see [1], [2]. In the following lemma, we see how we can distinguish between pairs of geometric hyperplanes in one or in distinct P SU 6 (2)orbits on GH. Lemma 2.2 Let H be a geometric hyperplane in GH. Then there are 567 elements of GH intersecting H in 375 points, and 840 elements intersecting H in 351 points. These 1408 hyperplanes form the P SU 6 (2)orbit on GH containing H. The hyperplane H intersects of each other P SU 6 (2)orbit on GH 112 elements in 405 points and 1296 in 357 points. Proof. Let G be the stabilizer of H in P SU 6 (2), and let q be a quad of H intersecting H in a subquadrangle of order (2, 2). Then the geometric hyperplane H tq intersects H in the 375 points of H that are at distance at most 1 from this subquadrangle. As we can find 567 such involutions, this yields a Gorbit of length 567. As each point of H is in /891 geometric hyperplanes and P SU 6 (2) has permutation rank 3 in its action on the cosets of O6 (3), the remaining 840 geometric hyperplanes form a Gorbit, and each of them intersects H in 351 vertices. (Count the number of incident vertexhyperplane pairs.) As follows from [2], there is a hyperplane H GH that meets H in 405 points, and the stabilizer of H in G is a group isomorphic to 3 4 : S 6. Hence G has an orbit of length 112 on these geometric hyperplanes. It follows from the information given in the Atlas [5], that G is transitive on the 1296 elements remaining in the P SU 6 (2)orbit of H. The elements from this Gorbit intersect H in 357 vertices. (Again this follows by counting incident pointhyperplane pairs.) 3. Proof of the Theorem Let Γ be a graph as given in the statement of Theorem 1.1. We shall prove in this section that the graph Γ is isomorphic to the graph G. Let a and b be two adjacent vertices of Γ. Then a and b have 378 common neighbors. If c is a vertex adjacent to both a and b, then c and b are collinear in the near hexagon induced on Γ a, if and only if they have 121 common neigbors in Γ a. They are at distance 2 in that 3
4 near hexagon, if they have 185 common neighbors. In particular, if b and c are collinear in the near hexagon induced on Γ a, then a and c are collinear in the near hexagon induced on Γ b. Let C be the collection of 4cliques in Γ with the following property. A 4clique C is in C, if and only if, for each vertex a in C, the three remaining vertices of C are a line of the near hexagon induced on Γ a. The above remarks show that C is nonempty, and that Γ together with the set C is an extended near hexagon in the terminology of [4]. Let a and c be two vertices of Γ that are at distance 2. Then by µ(a, c) we denote the subgraph of Γ induced on the set of common neighbors of a and c. Proposition 3.1 Let a and c be two vertices of Γ at distance 2. Then Γ a \ µ(a, c) is a geometric hyperplane of the near hexagon induced on Γ a, this hyperplane is in GH. Proof. Suppose a, b, c is a path of length 2 in the graph Γ. Then in the near hexagon on Γ b we see that a and c are at distance 3, and that c is at distance 2 and hence adjacent to a unique point on each of the lines of this near hexagon through a. Thus in the near hexagon induced on Γ a, the vertex c is adjacent to exactly two points on the lines through b. As b was arbitrarily chosen in the set of common neighbors of a and c, we see that c is adjacent to either 0 or 2 points of a line in the near hexagon on Γ a. Thus Γ a \ µ(a, c) is a geometric hyperplane of that hexagon. Suppose now that {a, b, d, e} C, and d is the unique vertex in this clique different from a and not adjacent to c. A vertex f in Γ b but different from a, d and e that is collinear to e in the near hexagon on Γ b, is adjacent to a, b, d and e. So in the near hexagon on Γ a it is at distance at most 2 from all the three points b, d and e that are on a line. As f is not collinear to a in Γ b, it is not collinear with b in Γ a. In Γ e we find that f is collinear with b, but not a, and thus f and e are not collinear in Γ a. Hence f has to be collinear with d inside the near hexagon on Γ a. Inside Γ b we see that there are 30 vertices f collinear with e that are not adjacent and 10 that are adjacent to c. So in Γ a we see that the vertex d is on 30/2=15 lines inside the geometric hyperplane Γ a \ µ(a, c) and 6 lines outside that hyperplane. So the proposition follows by the results of the previous section. Corollary 3.2 The graph Γ is strongly regular with parameters (v, k, λ, µ) = (2300, 891, 378, 324). Proof. Since the hyperplanes in GH have no deep points, we see that the graph Γ has diameter 2. The rest is now trivial. Corollary 3.3 Let a be a vertex of Γ, then the subgraph of Γ induced on the vertices distinct and nonadjacent to a, is a graph which is locally the distance 1 or 2 graph of the Aschbacher near hexagon. Proof. Straightforward. 4
5 Let a be a vertex of Γ, and for each vertex b nonadjacent to a denote by H b the geometric hyperplane of the near hexagon induced on Γ a consisting of all vertices adjacent to a, but not to b. Suppose b and c are two vertices different from a and nonadjacent to a. Since the two hyperplanes H b and H c cannot cover the whole Γ a, there is a vertex d adjacent to a, b and c. Inside Γ d we find a vertex e H b \ H c, and in particular we have that H b H c. Furthermore, we see that there is a vertex adjacent to b and c, but not to a. If b and c are adjacent, then they have 246 common neighbors that are not adjacent to a, and hence =132 common neighbors in Γ a. Thus H b H c contains =375 vertices, and H b and H c are in the same P SU 6 (2)orbit on GH. Fix one of the three P SU 6 (2)orbits on GH, containing H b. (Since Out(P SU 6 (2)) is transitive on the three orbits, it is of no concern, which of the three we take.) From the above we can deduce that we can identify each of the 1408 vertices c of Γ that are not adjacent to a with the unique geometric hyperplane H c in this orbit, and that for each geometric hyperplane H in this orbit there is a unique vertex c of Γ not adjacent to a with H c = H. As follows from Lemma 2.2, two vertices b and c of Γ, not adjacent to a are adjacent, if and only if H b H c = 375. This shows us that the graph Γ is unique up to isomorphism, and we have proven Theorem 1.1. References 1. M. Aschbacher, Flag structures on Tits geometries, Geom. Dedicata 14 (1983), A.E. Brouwer, A.M. Cohen, J.I. Hall and H.A. Wilbrink, Near polygons with lines of size three and Fischer spaces, in preparation. 3. F. Buekenhout, Diagram geometries for sporadic groups, in Finite groups coming of age, Contemp. Math. 45 (1985), 1 32, AMS, Providence. 4. P. Cameron, D. Hughes and A. Pasini, Extended generalised quadrangles, Geom. Dedicata 35 (1990), J. H. Conway et al., Atlas of finite groups, Clarendon Press, Oxford, Th. Meixner, A geometric characterization of the simple group Co 2, to appear in J. of Algebra. 7. A. Pasini and S. Yoshiara, Flagtransitive Buekenhout geometries, to appear in Reports of Gaeta conference, S. Yoshiara, On some extended dual polar spaces I, preprint. 5
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