A (c. L* )-Geometry for the Sporadic Group J2

Geometriae Dedicata 73: 225–235, 1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands.

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A (c · L∗)-Geometry for the Sporadic Group J2 FRANCIS BUEKENHOUT and CÉCILE HUYBRECHTS? Université Libre de Bruxelles, Service de Géometrie, CP 216, Campus Plaine, Bd du Triomphe, B-1050 Bruxelles, Belgique; e-mail: [email protected] [email protected] (Received: 24 December 1996) Abstract. We prove the existence of a rank three geometry admitting the Hall–Janko group J2 as flag-transitive automorphism group and Aut(J2 ) as full automorphism group. This geometry belongs to the diagram (c · L∗ ) and its nontrivial residues are complete graphs of size 10 and dual Hermitian unitals of order 3. Mathematics Subject Classifications (1991): 51E24, 20D08. Key words: diagram geometry, groups and geometries, Hall–Janko group J2 .

1. Introduction We prove the existence of a (c · L∗ )-geometry admitting the sporadic simple group J2 as flag-transitive automorphism group. The geometry we describe here was obtained by M. Hermand in unpublished work [12] supervised by F. Buekenhout. Hence, we call it the ‘Buekenhout–Hermand geometry’. It is mentioned in the Buekenhout ‘catalogue’ [3, Item (104)]. We were recently reminded by J. I. Hall that in 1984 he observed and discussed with the first author a concrete realization of this geometry within the representation discussed in [11]. The Buekenhout–Hermand geometry is a rank three incidence structure of points, lines and planes. It can be constructed as follows from the representation of degree 100 of the Hall–Janko group J2 . The points and planes of the geometry are the maximal subgroups of J2 of respective indices 100 and 280. Incidence between points and planes is defined in a natural way by observing that the index 280 subgroups may be identified with some sets of size 10 of index 100 subgroups. The lines of the geometry are defined as pairs of points contained in some plane. Here is a summary of the results established in the present paper. The Buekenhout–Hermand geometry is a flag-transitive geometry over the diagram

c · L∗ the full automorphism group of which is Aut(J2 ). Its flag-transitive automor-

phism groups are J2 and Aut(J2 ). Its point residues are dual Hermitian unitals of order 3. ? Charg´e de Recherches du Fonds National Belge de la Recherche Scientifique.

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The collinearity graph of the Buekenhout–Hermand geometry is a strongly regular graph of parameters (100, 63, 38, 42). It is one of the natural graphs associated to the representation of degree 100 of Hall–Janko group J2 . That graph was discovered by M. Hall, Jr and D. B. Wales [9]. Later on, J. Tits [19] constructed that graph from a generalized hexagon in order to give a computer free proof of the existence of the Hall–Janko group J2 . We also observe that J. Tits [20] gave a simpler interpretation of J2 as automorphism group of a quaternion version of the Leech lattice. For quite a number of years, geometers in Brussels tried unsuccessfully to generalize the construction of the Buekenhout–Hermand geometry on a purely combinatorial basis. Recently, the second author and A. Pasini [14] showed that the Buekenhout–Hermand geometry is the only flag-transitive (c · L∗ )-geometry whose point residues are dual unitals, thus explaining its isolation to some extent. As a consequence, this geometry is more important than it might have looked like in the past and proving its existence becomes a necessity. We now give a sketch of the paper. The construction of the Buekenhout– Hermand geometry 0 is given in Section 3. Next, in Section 4, we show that the group Aut(J2 ) preserves the structure of the geometry and that its subgroup J2 acts flag-transitively on 0. In Section 5, we determine the full automorphism group of 0 as well as its flag-transitive subgroups, we give a diagram for 0, we show that the point residues of 0 are dual Hermitian unitals of order 3; moreover we determine the structure of some flag stabilizers. Finally, in Section 6 we gather some useful information on the geometry and give a map of it as well as some of its properties and a Boolean lattice of flag stabilizers in J2 .

2. Preliminaries We assume some knowledge of basic facts from the theory of diagram geometries (see, for instance, Buekenhout [4, 2] or Pasini [18]) and on permutation groups (see, for instance, Wielandt [21] or Dixon–Mortimer [7]). We now recall some definitions and notation.

2.1.

LINEAR SPACES AND (c · L∗ )- GEOMETRIES

We recall that a linear space is a rank 2 geometry consisting of points and lines such that any two points are incident with exactly one line, any point is incident with at least two lines and any line with at least two points. A circle is a linear space with exactly two points on each line. The dual L∗ of a rank two geometry L of points and lines is the geometry obtained from L by permuting the role of points and lines. A (c · L∗ )-geometry is a rank three geometry of points, lines and planes, the plane, line and point residues of which are, respectively, circles, generalized digons

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and dual linear spaces (that is, the plane and line sets of every point residue are, respectively, the point and line sets of some linear space). We say that a geometry is flag-transitive if it admits a flag-transitive automorphism group.

2.2.

STABILIZERS AND KERNELS

Let 0 be a geometry and let G be an automorphism group of 0. We denote by GF the stabilizer in G of a flag F of 0 and by KF (G, 0) (or simply KF (G)) the kernel of GF for 0, that is the subgroup of GF fixing 0F elementwise. The stabilizer in G of a maximal flag of 0 is called a Borel subgroup of G and is denoted by B(G).

2.3.

HERMITIAN UNITALS

The Hermitian unital UH (q) of order q is a linear space usually defined from a nondegenerate Hermitian polarity of PG(2, q 2 ) (see, for instance, Hughes–Piper [13]). The one of order 3 can also be described as follows. LEMMA 2.1. The Hermitian unital of order 3 is the unique linear space having 28 points, 63 lines, 4 points on each line and the property that U3 (3) acts flagtransitively on it with a line stabiliser isomorphic to 4 · S4 . Proof. It is easy to see that the Hermitian unital of order 3 satisfies the properties of the statement. Therefore, to end the proof, it suffices to show the uniqueness of a linear space L with these properties. It is well-known that U3 (3) has a unique transitive action of degree 28. Let (U3 (3), ) be such an action, where  is a set of size 28. Since L is a linear space, the lines of L are uniquely determined by their point sets. Let L be a line of L, and let Q be its stabilizer in U3 (3). By assumption, Q is isomorphic to 4 · S4 and is transitive on the four points of L. Therefore, Q has at least two orbits on , including one of size 4. However, there is only one conjugacy class of subgroups isomorphic to 4 · S4 in U3 (3) and such subgroups have two orbits on . As a consequence, these orbits are of respective sizes 4 and 24. We may then reconstruct the lines of L by taking as lines the orbits of length 4 on  of the 2 subgroups isomorphic to 4 · S4 .

3. Construction In this section, we construct the Buekenhout–Hermand geometry. To this end, we first gather some information on the Hall–Janko group J2 and we define some objects, called ‘10-varieties’, that will be the planes of the geometry.

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THE HALL – JANKO GROUP J2

In the present paper, we freely use information on the Hall–Janko group J2 and its subgroups that may be found in the Atlas [6] of J. Conway, R. Curtis, S. Norton, R. Parker and R. Wilson and we follow their notation. For constructions or more information on J2 , we refer to Janko [15, 16], Hall and Wales [9, 10], Tits [19, 20] and Gorenstein [8, §2.4 and §2.6] (see also Aschbacher [1] for a unified theory of sporadic groups). We recall that the Hall–Janko group J2 has exactly one conjugacy class of subgroups of index 100 (resp. 280), these subgroups being isomorphic to U3 (3) (resp. 3 · PGL2 (9)). As a consequence, J2 has a unique faithful transitive action of degree 100. Let (J2 , S) be such an action, where S is a set of size 100. From the character table of J2 , we easily deduce that J2 has exactly two conjugacy classes C1 and C2 of elements of order 3, with a centralizer of respective sizes 1080 and 36, the elements of C1 (resp. C2 ) fixing exactly 10 (resp. 4) elements of S. Let C denote the conjugacy class of subgroups of index 280 in J2 . Clearly, each member H of C has a unique normal subgroup of order three. We call it the normal 3-subgroup of H . By its maximality in J2 , any member of C is in fact the normalizer in J2 of its normal 3-subgroup. The generators of the latter are members of C1 (see the ‘abstract specifications’ of H given in [6]). 3.2. 10- VARIETIES For every subset A of J2 , we denote by Fix(A) the set of elements of S fixed under A, that is by every element of A. LEMMA 3.1. Every member H of C has exactly two orbits on S, of respective lengths 10 and 90. The one of size 10 is the set of elements fixed by the normal 3-subgroup of H . Proof. Let N be the normal 3-subgroup of H and let n be a generator of N. Clearly, Fix(N) = Fix(n) and this set is of size 10 since n is a member of C1 . Moreover, by the normality of N, the set Fix(N) is stabilized by H . On the other hand, using the permutation characters of J2 provided by the Atlas [6], it is easy to see that every member H of C has two orbits on S. Therefore, the 2 two orbits of H on S are exactly Fix(N) and S \ Fix(N). For each member H of C, we denote by O10 (H ) the orbit of length 10 of H on S. A subset O of S is called 10-variety of S if there is a member H of C such that O = O10 (H ). By the preceding lemma, a 10-variety may also be defined as the set of fixed elements of the normal 3-subgroup of some member of C. In other words, a set O is a 10-variety if and only if there is a group N generated by a member of C1 such that O = Fix(N).

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0

LEMMA 3.2. For every element x of S, there exists a 10-variety containing x. Proof. The stabilizer of x in J2 is isomorphic to U3 (3). The latter has two conjugacy classes of elements of order 3, with centralizer of respective sizes 108 and 9. As a consequence, (J2 )x contains elements of the class C1 of J2 (namely those whose centralizer in U3 (3) is of size 108). Let N be the group generated by such 2 an element. In this case, by definition, Fix(N) is a 10-variety containing x. DEFINITION. The Buekenhout–Hermand geometry 0 is defined by taking the elements of S as points, the 10-varieties as planes, the pairs of points contained in some 10-variety as lines, the incidence being symmetrized inclusion. By Lemma 3.2, it is straightforward to see that 0 is indeed a geometry. 4. Flag-Transitive Automorphism Groups of 0 LEMMA 4.1. The group Aut(J2 ) is an automorphism group of 0. Proof. The group Aut(J2 ) preserves the set of 10-varieties of 0 since it stabilizes 2 the conjugacy class C of J2 . The result immediately follows. LEMMA 4.2. There are 280 planes in 0. Moreover, J2 acts transitively on them, with a plane stabilizer isomorphic to 3 · PGL2 (9). Proof. Let O be a 10-variety of S. Then by definition, there is a member H of C such that O = O10 (H ). In particular, H stabilizes O. By the maximality of H in J2 and by the transitivity of the permutation group (J2 , S), every element of J2 stabilizing O is in H . As a consequence, the stabilizer of O in J2 is precisely H and O is not stabilized by any other member of C. Therefore the 10-varieties uniquely correspond to the members of C. However, by definition of a conjugacy class, J2 acts transitively on the latter. Moreover C is of size 280 (for indeed, the size of a conjugacy class of maximal subgroups is equal to the index of its members in the 2 group). The statements immediately follow. LEMMA 4.3. A plane stabilizer acts 3-transitively on the point set of the plane, with a kernel equal to its normal 3-subgroup. Proof. Let π be a plane of 0. By Lemma 4.2, the stabilizer of π in J2 is a member H of C. Let K be the kernel of H and let N be the normal 3-subgroup of H . By Lemma 3.1, N is included in K. We now show that these sets are equal. By construction, H is transitive on the point set of π . Hence, for every point p of π , the stabilizer Hp is of index 10 in H . However, K is included in such stabilizers. Therefore K is of index at least 10 in H . On the other hand, K/N is a normal subgroup of H/N which is isomorphic to the almost simple group PGL2 (9). As a consequence, K/N is either reduced to the identity or of index at most two in H/N, and so K is either equal to N or of index at most two in H . However, the latter possibility is excluded by the above.

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Since K = N, the group H induces PGL2 (9) on the residue of π . Moreover it is well-known (see, for instance, [6]) that PGL2 (9) has a unique action of degree 10, this action being 3-transitive. 2 From Lemmas 4.1, 4.2 and 4.3, we immediately deduce the following result. PROPOSITION 4.4. The groups J2 and Aut(J2 ) are flag-transitive automorphism groups of 0. 5. Diagram and Automorphism Groups of 0 5.1.

COLLINEARITY GRAPH AND FLAG - TRANSITIVE AUTOMORPHISM GROUPS OF 0

LEMMA 5.1. For each point p of 0, the stabilizer (J2 )p has three orbits on the point set of 0, of respective lengths 1, 63 and 36, these sets being, respectively, the set of points of 0 at distance zero, one and two from p. Moreover, for each line L of 0 through p,the group (J2 )L (resp. (J2 )p,L ) has structure 4 · S4 · 2 (resp. 4 · S4 ). Proof. By construction, (J2 )p is isomorphic to U3 (3). By the maximality of U3 (3) in J2 , the stabilizer (J2 )p is its own normalizer in J2 , and so it does not fix any other point of 0. Therefore the size of any of its orbits on the 99 points of 0 \ {p} is a multiple of 28, 36 or 63. It is easy to deduce that there is only one possibility for the orbits of (J2 )p on the point set of 0 \ {p}, namely two orbits of respective lengths 36 and 63. Since J2 acts flag-transitively on 0, the number k of points collinear with p must be either 36 or 63. We saw in Lemma 3.2 that (J2 )p has an element of order 3 fixing 9 points collinear with p (namely a generator n of the pointwise stabilizer of a 10-variety containing p) and that this element has a centralizer in (J2 )p of size 108. However, using a character table of U3 (3), it is easy to see that an element of order 3 of U3 (3) with a centralizer of size 108 fixes 9 elements only for the action of U3 (3) of degree 63 with 4 · S4 as element stabilizer. As a consequence, k = 63 and (J2 )p,L = 4 · S4 for any line L through p. Since (J2 )p,L is of index 2 in (J2 )L , we have that (J2 )L = 4 · S4 · 2. It is straightforward to see that the orbit of length 36 is exactly the set of points 2 at distance two from p. By the above lemma, the collinearity graph of the Buekenhout–Hermand geometry 0 is a graph of degree 63 on 100 vertices on which J2 acts nontrivially. However it is well-known that such a graph is uniquely determined. As a consequence, it is the strongly regular graph of parameters (100, 63, 38, 42) defined in Tits [19], with Aut(J2 ) as full automorphism group. PROPOSITION 5.2. We have Aut(0) = Aut(J2 ) = J2 · 2. Moreover, the only flag-transitive automorphism groups of 0 are J2 and Aut(J2 ).

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Proof. By Lemma 4.1, Aut(J2 ) is an automorphism group of 0. On the other hand, the full automorphism group of 0 is contained in the automorphism group of its collinearity graph, which is Aut(J2 ). Therefore, Aut(0) = Aut(J2 ). By Proposition 4.4, J2 and Aut(J2 ) are flag-transitive automorphism groups of 0. Moreover, using the Atlas [6], it is easy to see that no other subgroup of Aut(J2 ) acts transitively on the couples of adjacent vertices of the collinearity graph 2 of 0.

5.2.

A DIAGRAM FOR

0

In this section, we show that 0 is a (c · L∗ )-geometry. LEMMA 5.3. There are 28 (resp. 4) planes of 0 through each point (resp. line) of 0. Proof. By construction, there are 100 points in 0. Moreover, by Lemmas 4.2 and 5.1, there are 280 planes in 0 and 63 lines through each point of 0. Since J2 acts flag-transitively on 0, the numbers of the statement do not depend on the chosen point or line. Therefore, the result can be obtained by counting in two different ways the number of flags of type {point, plane} in 0 and the number of flags of 2 type {line, plane} in the residue of a given point. LEMMA 5.4. Any two nondisjoint planes of 0 have exactly two common points. In other words, the point residues of 0 are dual linear spaces. Proof. Assume by way of contradiction that there are two planes α and β1 of 0 having three points in common. Let a and b be two of them. Let H be the stabilizer in J2 of α. By Lemma 5.3, there are exactly two planes β2 and β3 through a and b that are distinct from α and β1 . Hence, the set {β1 , β2 , β3 } is stabilized by Ha,b . We now show that any intersection α ∩ βi which is stabilized by Ha,b is necessarily of size two. By Lemma 4.3, Ha,b is transitive on the point set of α\{a, b}. As a consequence, the intersection α ∩ βi is either of size two or equal to α. However the latter possibility is excluded by definition of the 10-varieties. Let N be the subgroup of H fixing α pointwise. We recall that N is a cyclic group of order 3 which is normal in H . Since N fixes every point of α, it stabilizes in particular each of the intersections α ∩ βi . On this basis, we now show that N stabilizes each of the βi . If this is not the case, then N permutes the βi transitively, and so the intersections α ∩ βi are pairwise equal. In particular, the intersection α ∩ β1 is stabilized by Ha,b . However, by our first step this is impossible since β1 intersects α in at least three points. We have just seen that N stabilizes each of the βi . As a consequence, each of the βi intersects α in at least three points, for indeed Fix(N) = α, so that each orbit of N on the set βi \α is of size three, and so βi \α is of cardinality multiple of three; in particular, α ∩ βi cannot be of size two.

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The group Ha,b is of size 24 and it contains N as a normal subgroup. Therefore every odd order element of Ha,b is included in N. Moreover, N stabilizes each of the βi . Therefore the orbits of Ha,b on the set {β1 , β2 , β3 } have size one or two, so that Ha,b stabilizes at least one of the βi . In particular, Ha,b stabilizes its intersection with α. However, this intersection is of size at least three, which contradicts our first step. Therefore any two planes of the geometry intersect in at most two points. Let δ be a plane of 0 and let p be one of its points. There are nine lines in δ through p. By Lemma 5.3, each of them is contained in three other planes. Since any two planes of 0 have at most two points in common, we obtain, in this way, 27 pairwise distinct planes through p intersecting δ in one line. On the other hand, by Lemma 5.3, this is exactly the number of planes through p and distinct from δ, 2 and so any two planes through p intersect in one line. As an easy corollary, the following holds. PROPOSITION 5.5. 0 is a (c · L∗ )-geometry. 5.3.

POINT RESIDUES OF

0

We have just seen that the point residues of 0 are dual linear spaces. Let us show that they are dual Hermitian unitals of order 3. We also determine some kernels and the Borel subgroup of the geometry. PROPOSITION 5.6. The point residues of 0 are dual Hermitian unitals of order 3. Moreover, the kernel of the point (resp. line) stabilizer in J2 is trivial (resp. cyclic of order 4) and B(J2 ) ' 3 · 8. Proof. Since the point stabilizers in J2 are isomorphic to the simple group U3 (3), their kernel is trivial, and so they induce on the point residue a group isomorphic to U3 (3). By Proposition 4.4, this action is flag-transitive. Moreover, by Lemma 5.1, the {point, line}-stabilizers in J2 are isomorphic to 4 · S4 . Therefore by Lemma 2.1, the point residues of 0 are dual Hermitianunitals of order 3. Let L be a line of 0 and let p be a point of L. Since the lines of 0 have ‘size’ two, KL (J2 , 0) = KL,p (J2 , 0). The latter is equal to KL ((J2 )p , 0p ) which is a cyclic group of order 4. (Indeed, any element of U3 (3) fixing a line L of the unital as well as its four absolute points also fixes the nonabsolute points of PG(2,9) that are on L; in other words, it is a perspectivity, and so KL (U3 (3), UH (3)) is a cyclic group of order 4.) The Borel subgroup of J2 for 0 is equal to the Borel subgroup 2 of U3 (3) for the unital, that is to 3 · 8.

6. More Information on the Geometry In this section, we provide a map of the Buekenhout–Hermand geometry and a Boolean lattice of flag stabilizers in J2 . We do not recall all conventions for them:

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Figure 1.

we refer to the Atlas [5] of F. Buekenhout, M. Dehon and D. Leemans for more details.

6.1.

A MAP OF THE GEOMETRY

In Figure 1, we give a distance distribution map of the Buekenhout–Hermand geometry developed from a point p (with the convention that every circle, square and hexagon is a box representing respectively a set of points, lines and planes of 0, the number written into a box being the cardinality of the corresponding set). Moreover, it can be shown that these boxes are in fact the orbits of the stabilizer (J2 )p on the elements of 0. Thus (J2 )p has, respectively, three, five and two orbits on the points, lines and planes of 0. Most of this information can be derived from the general analysis carried on in Huybrechts and Pasini [14] about (c · L∗ )-geometries with dual unitals as point residues. One of the purposes of the map is to allow further comparisons with buildings. We observe, for instance, a property that reminds us of polar spaces: given a point p and a plane π not incident with p, there is a constant number (six here) of points of π that are collinear with p.

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Figure 2.

6.2.

BOOLEAN LATTICE OF STABILIZERS IN J2

In Figure 2, we give the Boolean lattice of flag stabilizers in J2 for 0. It has four levels (we count them from top to bottom). At the ith level, where i ∈ {0, 1, 2, 3}, we put the conjugacy class of the stabilizer of a flag of 0 of size i. We surround by a square the kernel of the action of this stabilizer on the corresponding flag residue. The information on the picture has been obtained on the basis of the present paper as well as the subgroup lattice of J2 provided by Pahlings [17]. Moreover, the following observations can also be derived from [17]. (i) Up to a conjugacy class of subgroups isomorphic to 21+4 : A5 , which are intermediates of line stabilizers and J2 , every subgroup of J2 containing a Borel subgroup of J2 for 0 is represented on Figure 2. In particular, except for the line stabilizers which are not maximal in J2 , all inclusions occurring on the lattice are maximal. (ii) Any Borel subgroup is contained into two {point, plane}-stabilizers. Up to this exception, the way to ‘go up’ from the Borel subgroups to the group J2 is uniquely determined by the arrows of the lattice.

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FURTHER REMARKS

The Buekenhout–Hermand geometry satisfies the intersection property (I P). Moreover, by the uniqueness result of Huybrechts–Pasini [14], it is simply connected. From Figure 2, it immediately follows that this geometry is RWPRI but not PRI in the sense of [5]. Another property is that the stabilizer of a comaximal flag is 2-transitive on the residue of the given flag. References 1.

Aschbacher, M.: Sporadic Groups, Cambridge Tracts in Math. 104, Cambridge Univ. Press, 1994. 2. Buekenhout, F.: The basic diagram of a geometry, in: M. Aiguer and D. Jungnickel (eds), Geometries and Groups, Lecture Notes in Math. 893, Springer-Verlag, Berlin, 1981, pp. 1–29. 3. Buekenhout, F.: Diagram geometries for sporadic groups, Contemp. Math. 45 (1985), 1–45. 4. Buekenhout, F.: Foundations of incidence geometry, in: F. Buekenhout (ed.), Handbook of Incidence Geometry, Elsevier, Amsterdam, 1995, pp. 63–105. 5. Buekenhout, F., Dehon, M. and Leemans, D.: An atlas of residually weakly primitive geometries for small groups, Mém. Soc. Math. Belg., To appear. 6. Conway, J., Curtis, R., Norton, S., Parker, R. and Wilson, R.: Atlas of Finite Groups, Clarendon Press, Oxford, 1985. 7. Dixon, J. D. and Mortimer, B.: Permutation Groups, Springer-Verlag, New York, 1996. 8. Gorenstein, D.: Finite simple groups, an introduction to their classification, Plenum, New York, 1982. 9. Hall, M. Jr. and Wales, D. B.: The simple group of order 604,800, J. Algebra 9 (1968), 417–450. 10. Hall, M. Jr. and Wales, D. B.: The simple group of order 604,800, in: R. Brauer and Chih-Han Sah (eds), Theory of Finite Groups, a Symposium, Benjamin, New York, 1969, pp. 79–90. 11. Hall, J. I. and Hall, M. Jr.: Geometry of the Hall–Janko group, Algebras, Groups Geom. 2 (1985), 390–398. 12. Hermand, M.: Du groupe de Hall–Janko aux semidesigns réguliers, Mémoire de licence, Université Libre de Bruxelles, 1982. 13. Hughes, D. R. and Piper, F. C.: Projective Planes, Springer-Verlag, New York, 1973. 14. Huybrechts, C. and Pasini, A.: A characterization of the Hall–Janko group J2 by a c · L∗ geometry, in: L. Di Martino et al. (eds), Groups and Geometries, Birkhäuser-Verlag, Basel, 1998, pp. 91–106. 15. Janko, Z.: Some new simple groups of finite order . I ., Sympos. Math. (INDAM), Rome (1967– 68), vol. I, pp. 25–64. 16. Janko, Z.: Some new simple groups of finite order, in: R. Brauer and Chih-Han Sah (eds), Theory of Finite Groups, a Symposium, Benjamin, New York, 1969, pp. 63–64. 17. Pahlings, H.: The subgroup structure of the Hall–Janko group J2 , Bayreuth. Math. Schr. 23 (1987), 135–165. 18. Pasini, A.: Diagram Geometries, Oxford Univ. Press, Oxford, 1994. 19. Tits, T.: Le groupe de Janko d’ordre 604,800, in: R. Brauer and Chih-Han Sah (eds), Theory of Finite Groups, a Symposium, Benjamin, New York, 1969, pp. 91–95. √ 20. Tits, J.: Quaternions over Q( 5), Leech’s lattice and the sporadic group of Hall–Janko, J. Algebra 62 (1980), 56–75. 21. Wielandt, H.: Finite Permutation Groups, Academic Press, New York, 1964.

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