# 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 flag-transitively. 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

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 3-transpositions, 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 G-orbit 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 G-orbit, and each of them intersects H in 351 vertices. (Count the number of incident vertex-hyperplane 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 G-orbit intersect H in 357 vertices. (Again this follows by counting incident point-hyperplane 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 4-cliques in Γ with the following property. A 4-clique 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

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