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The Subgroup Structure of the Mathieu Group Mi2 MATHEMATICS OF COMPUTATION VOLUME 50, NUMBER 182 APRIL 1988, PAGES 595-605 The Subgroup Structure of the Mathieu Group Mi2 By Francis Buekenhout and Sarah Rees Abstract. We present a full description of the lattice of subgroups of the Mathieu group Mi2. 1. Introduction. The recently fast developing search for geometries invariant by a finite simple group G (see [1], [9], [7], [2]) makes it suitable to dispose of various portions of the subgroup lattice of G. Hence it may be interesting to determine once and for all the full subgroup lattice of G. Of course, this lattice may be too large for a useful description. It seems more realistic to look for the subgroup pattern, in which the elements are conjugacy classes of subgroups and order is provided by inclusion. Even so, the task may look too difficult for any of the sporadic groups, but the result is surprisingly elegant for groups such as Mu and Ji (see [2]) while the method of search is obviously unpleasant. Large portions of the subgroup pattern of the sporadic group Mc have been worked out by O. Diawara [5]. The purpose of the present paper is to discuss the subgroup pattern of the Mathieu group Mi2. In calculating this, some use was made of the information available in [4], [8] and [6] and of the Cayley programming language ([3]). 2. Results. It turns out that there are 147 conjugacy classes of subgroups in Mi2- We denote these classes by Ki,K2,... and list them in increasing order, somewhat as Cayley does. We take the outer automorphism group of Mi2, namely M12.2, into account as follows. When two classes of Mi2 are fused under Mi2-2, we denote them by Ki and Ki. In view of this twinning procedure, the labels i run from 1 to 107 instead of 147. If no confusion can arise, the class Kx may be called "class x". For each class Kx, we give one or more descriptions of the isomorphism type of its members, using fairly standard notation for this, in particular, the conventions of the Atlas [4]. So "6" stands for a cyclic group of order six, D§ for a dihedral group of order 8, AB for a group G having a normal subgroup of type A with G/A ~ B, etc. We list (see table in Appendix) the classes Kx, the length or cardinality of Kx, the class Ky of the normalizer of a member of Kx, the number of maximal subgroups and of minimal overgroups in each possible class for a given member of Kx. Since Mi2 has a particularly interesting representation on the 12 left cosets of some subgroup Mi 1, we give the lengths of the orbits of a member H of Kx on these 12 points by a notation such as [4,2,2,2], [4 x 3] or [26]. These mean that Received January 17, 1985; revised May 27, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 20D30. ©1988 American Mathematical Society 0025-5718/88 $1.00 + $.25 per page 595 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 596 FRANCIS BUEKENHOUT AND SARAH REES H has an orbit of 4 points and 4 orbits of 2 points in the first case, that H has an orbit of 12 points with 2 systems of imprimitivity blocks of sizes 4 and 3 respectively in the second case and that H has an orbit of 12 points with 6 blocks of imprimitivity of size 2 in the third case. Département de Mathématique Campus Plaine C.P. 216 Université Libre de Bruxelles 1050 Bruxelles, Belgium Department of Mathematics University of Illinois at Chicago Box 4348 Chicago, Illinois 60680 1. F. BUEKENHOUT, "Diagram geometries for sporadic groups," in Finite Groups—Coming of Age, Conf. Proceedings (J. McKay, ed.), Contemporary Mathematics, vol. 45, Amer. Math. Soc, Providence, RI, 1985. 2. F. BUEKENHOUT, "The geometry of the finite simple groups," in Buildings and the Geometry of Diagrams, Como, 1984 (L. Rosati, ed.), Lecture Notes in Math., vol. 1181, Springer-Verlag, Berlin, Heidelberg, New York. 3. J. CANNON, A Language for Group Theory, Univ. of Sydney, 1982. 4. J. H. CONWAY, R. T. CURTIS, S. P. NORTON, R. A. PARKER & R. A. WILSON, Atlas of Finite Groups, Oxford Univ. Press, 1985. 5. O. DlAWARA, Sur le Groupe Simple de J. McLaughlin, Thesis, Université Libre de Bruxelles, 1984. 6. V. FELSCH & G. SandlÖBER, "An interactive program for computing subgroups," Com- putational Group Theory (M. Atkinson, ed.), Academic Press, New York, 1984. 7. M. RONAN & G. STROTH, "Minimal parabolic geometries for the sporadic groups," European J. Combin.,v. 5, 1984, pp. 59-92. 8. A. D. THOMAS & G. V. WOOD, Group Tables, Shiva Publ. Limited, 1980. 9. J. TITS, "Buildings and Buekenhout geometries," in Finite Simple Groups II (M. Collins, ed.), Academic Press, New York, 1981, pp. 309-320. Appendix: Subgroups of Mi2. See table on the following pages. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use THE SUBGROUP STRUCTURE OF THE MATHIEU GROUP M,2 597 o CO CO I k Ol CO ICO o CO CM ICM IS CO k k k CM o CO CO CM CO oí k es k00 co 05 •* ICO CO CO IS CO is ira k k k * k o CO CM CM ,-< CO * s i-H 00 CO 00 ICO CO CO CO o CM Ici CO t- IS CO * 6«! k k •* k CM CO CO CM o CM O) Oí I« CN. lira cm o ira .H -H CO CM CN CN CO ira CO o k k k CO S ¡«i kCO CO ^H CO k CO 00 Ol 00 IO) ira o Oí lO rt CO •H ICO CO <N N CN en ■* Ico' CO SÇ ix;* so X ¡*! * ~H CO CO cm "*r CM Tí w CO CO —I CM cm' CO CO t- CM O CJ) ^h ira CM -H kCO 15IS O) k k x ¡*! * i*:* co -H CO 00 CO W GO ~H CO CO CO CO ira X I«! 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