"The close relationships between group theory and structural combinatorics go back well over a century. Given a combinatorial object, it is natural to consider its automorphism group. Conversely, given a group, there may be a nice object upon which it acts. If the group is given as a group of permutations of some set, it is natural to try to regard the elements of that set as the points of some structure which can be at least partially visualized. For example, in 1861 Mathieu... discovered five multiply transitive permutation groups. These were constructed as groups of permutations of 11, 12, 22, 23 or 24 points, by means of detailed calculations. In a little-known 1931 paper of Carmichael [5], they were first observed to be automorphism groups of exquisite finite geometries. This fact was rediscovered soon afterwards by Witt [11], who provided direct constructions for the groups and then the geometries. It is now more customary to construct first the designs, and then the groups...."   5.  R. D. Carmichael, Tactical configurations of rank two, Amer. J. Math. 53 (1931), 217-240. 11.  E. Witt, Die 5-fach transitiven Gruppen von Mathieu, Abh. Hamburg 12 (1938), 256-264. -- William M. Kantor, book review (pdf), Bulletin of the American Mathematical Society, September 1981

More details on Carmichael and Witt

(Like the above, these brief excerpts from
 copyrighted publications are quoted here
 for personal scholarly purposes only.)

Excerpt from Chapter III, "The Affine Group in the Plane"--

"The system of definitions and theorems which expose properties invariant under a given group of transformations may be called, in agreement with the point of view expounded in Klein's Erlangen Programm,* a geometry.  Obviously, all the theorems of the geometry corresponding to a given group continue to be theorems in the geometry corresponding to any subgroup of the given group; and the more restricted the group, the more figures will be distinct relatively to it, and the more theorems will appear in the geometry.  The extreme case is the group corresponding to the identity, the geometry of which is too large to be of consequence.

For our purposes we restrict attention to groups of projective collineations,** and in order to get a more exact classification of theorems we narrow the Kleinian definition by assigning to the geometry corresponding to a given group only the theory of those properties which, while invariant under this group, are not invariant under any other group of projective collineations containing it.  This will render the question definite as to whether a given theorem belongs to a given geometry.

Perhaps the simplest example of a subgroup of the projective group in a plane is the set of all projective collineations which leave a line of the plane invariant.  The present chapter is concerned chiefly with the geometry belonging to this group." 

* Cf. F. Klein, Vergleichende Betractungen uber neue geometrische Forschungen, Erlangen 1872; also in Mathematische Annalen, Vol. XLIII (1893), p. 63

** From some points of view it would have been desirable to include also all projective groups containing correlations.

The MOG may be regarded as summarizing a geometric structure featuring 759 eight-sets ("octads") in a "space" of 24 points.  The set of 759 octads is left invariant by the transformations of the large Mathieu group M₂₄.  A subgroup, the octad stabilizer, leaves an octad invariant (as a set, though not pointwise).  It so happens that the octad stabilizer, which may be viewed as acting separately on the stabilized octad and on the set of the remaining 16 points of the 24, is isomorphic to the automorphism group of the affine 4-space over the 2-element field.  (In fact, the two parts of this group's action illustrate the isomorphism of the alternating group on eight elements with the general linear group acting on the 4-space over the 2-element field; the stabilized octad is left fixed pointwise by the translations of the affine 4-space.)  A standard theorem of projective geometry says that we may adjoin to an affine n-space an (n-1)-dimensional projective "hyperplane at infinity" to obtain a projective n-space.  The group of the geometry of the affine n-space is the subgroup of the group of the projective n-space leaving the "hyperplane at infinity" invariant (as a set, not pointwise), as in Veblen's discussion above.  One might argue that a stabilized octad in the Mathieu geometry plays a role in some way analogous to that of a "hyperplane at infinity."

1. Chapter 5, "Sporadic Groups," of Finite Simple Groups, by Robert A. Wilson (in preparation as of August 15, 2007).
   
   
2. The Conway-Sloane book has a version of the MOG that differs from the original Curtis MOG by a mirror reflection:
   
   
   The first edition of the Conway-Sloane book was published in 1988. Conway and Sloane supply a brief discussion of the square structures in the Curtis MOG as pictures of the finite geometry AG(4,2), but make no reference to the work, ten years earlier, of Cullinane.
   
   
3. Steven H. Cullinane, Geometry of the 4x4 Square and The Diamond Theorem. The former gives details of how the MOG works. The latter gives details, based on a Feb. 1979 AMS Notices abstract, of how the 35 4x4 structures in the MOG may be viewed as the 35 lines in the projective 3-space over the 2-element field.
   
   
4. Steven H. Cullinane, Generating the Octad Generator. This 1985 note is related to Curtis's revised view of the MOG in a 1987 article. The following journal note by Cullinane points this  out:
   
   

   Monday, January 05, 2009 A Wealth of Algebraic Structure A 1987 article by R. T. Curtis on the geometry of his Miracle Octad Generator (MOG) as it relates to the geometry of the 4x4 square is now available online: Further elementary techniques using the miracle octad generator, by R. T. Curtis. Abstract: "In this paper we describe various techniques, some of which are already used by devotees of the art, which relate certain maximal subgroups of the Mathieu group M24, as seen in the MOG, to matrix groups over finite fields. We hope to bring out the wealth of algebraic structure* underlying the device and to enable the reader to move freely between these matrices and permutations. Perhaps the MOG was mis-named as simply an 'octad generator'; in this paper we intend to show that it is in reality a natural diagram of the binary Golay code." (Received July 20 1987) -- Proceedings of the Edinburgh Mathematical Society (Series 2) (1989), 32: 345-353, doi:10.1017/S0013091500004600. (Published online by Cambridge University Press 19 Dec 2008.) In the above article, Curtis explains how two-thirds of his 4x6 MOG array may be viewed as the 4x4 model of the four-dimensional affine space over GF(2).  (His earlier 1974 paper (below) defining the MOG discussed the 4x4 structure in a purely combinatorial, not geometric, way.) For further details, see Geometry of the 4x4 Square and Curtis's original 1974 article, which is now also available online: A new combinatorial approach to M24, by R. T. Curtis. Abstract: "In this paper, we define M24 from scratch as the subgroup of S24 preserving a Steiner system S(5, 8, 24). The Steiner system is produced and proved to be unique and the group emerges naturally with many of its properties apparent." (Received June 15 1974) -- Mathematical Proceedings of the Cambridge Philosophical Society (1976), 79: 25-42, doi:10.1017/S0305004100052075. (Published online by Cambridge University Press 24 Oct 2008.) * For instance: Click for details.

   
   
5. Symmetric Generation of Groups, by Robert T. Curtis, Encyclopedia of Mathematics and Its Applications Vol. 111, Cambridge University Press, 2007. This book, like the 1987 article by Curtis above, views the MOG in a wider context.


6. The connection of the Steiner system S(5, 8, 24) with finite geometry-- in particular, with PG(3,2) and with the correspondence in Conwell's 1910 paper-- is described in Wilbur Jonsson's "On the Mathieu Groups M₂₂, M₂₃, M₂₄ and the Uniqueness of the Associated Steiner Systems" (Math. Z. 125, 193-214 (1972)), received on 9 November 1970 by Mathematische Zeitschrift--
   
   

   Adapted (for HTML) from the opening paragraphs of the above paper on Steiner systems by Jonsson-- "[A]... uniqueness proof is offered here based upon a detailed knowledge of the geometric aspects of the elementary abelian group of order 16 together with a knowledge of the geometries associated with certain subgroups of its automorphism group. This construction was motivated by a question posed by D.R. Hughes and by the discussion Edge [5] (see also Conwell [4]) gives of certain isomorphisms between classical groups, namely PGL(4,2)~PSL(4,2)~SL(4,2)~A8, PSp(4,2)~Sp(4,2)~S6, where A8 is the alternating group on eight symbols, S6 the symmetric group on six symbols, Sp(4,2) and PSp(4,2) the symplectic and projective symplectic groups in four variables over the field GF(2) of two elements, [and] PGL, PSL and SL are the projective linear, projective special linear and special linear groups (see for example [7], Kapitel II). The symplectic group PSp(4,2) is the group of collineations of the three dimensional projective space PG(3,2) over GF(2) which commute with a fixed null polarity tau...." References 4. Conwell, George M.: The three space PG(3,2) and its group. Ann. of Math. (2) 11, 60-76 (1910). 5. Edge, W.L.: The geometry of the linear fractional group LF(4,2). Proc. London Math. Soc. (3) 4, 317-342 (1954). 7. Huppert, B.: Endliche Gruppen I. Berlin-Heidelberg-New York: Springer 1967.

   
   
7. Thomas M. Thompson, From Error-Correcting Codes through Sphere Packings to Simple Groups. Mathematical Association of America, Washington, 1983.