On geometries of the Conway group Co₃ and their McLaughlin group subgeometries

Abstract

Recently, the author published a book [A. A. Ivanov, Ever-Evolving Groups—an Introduction to Modern Finite Group Theory, Algebr. Appl. 32, Springer, Cham, 2025] where he summarised the recent progress in the geometric theory of sporadic groups and outlined some geometries which require further investigation. Among them was a geometry for the smallest Conway sporadic simple group Co 3 , with diagram originally introduced in [M. A. Ronan and G. Stroth, Minimal parabolic geometries for the sporadic groups, European J. Combin. 5 (1984), 1, 59–91] (cf. [M. A. Ronan, Coverings of certain finite geometries, Finite Geometries and Designs, London Math. Soc. Lecture Note Ser. 49, Cambridge University, Cambridge (1981), 316–331, Table 1] and [F. Buekenhout, Diagram geometries for sporadic groups, Finite Groups—Coming of Age (Montreal 1982), Contemp. Math. 45, American Mathematical Society, Providence (1985), geometry (23), p. 14]), which we denote by G ⁢ ( Co 3 ) . The 2-local geometries for the other Conway groups Co 1 and Co 2 are the tilde and Petersen geometries which have been intensively studied (cf. [A. A. Ivanov and S. V. Shpectorov, The flag-transitive tilde and Petersen-type geometries are all known, Bull. Amer. Math. Soc. (N. S.) 31 (1994), 2, 173–184]). However, G ⁢ ( Co 3 ) seems to be studied less. There is another geometry associated with Co 3 (cf. [M. A. Ronan, Coverings of certain finite geometries, Finite Geometries and Designs, London Math. Soc. Lecture Note Ser. 49, Cambridge University, Cambridge (1981), 316–331, Table 1] and [F. Buekenhout, Diagram geometries for sporadic groups, Finite Groups—Coming of Age (Montreal 1982), Contemp. Math. 45, American Mathematical Society, Providence (1985), geometry (23), p. 14]) with diagram The elements of type 1, 2, 3, and 4 in the geometry G 276 correspond to cliques of size 1, 2, 3, and 6, respectively, in a double cover of the complete graph on 276 vertices. The group Co 3 acts doubly transitively on the vertex set of the complete graph and flag-transitively on G 276 . This double cover is naturally associated with the well-known 2-graph of Co 3 . The geometry G 276 is simply connected, as established by [M. A. Ronan, Coverings of certain finite geometries, Finite Geometries and Designs, London Math. Soc. Lecture Note Ser. 49, Cambridge University, Cambridge (1981), 316–331, Proposition 6]. In the present paper, we recover the distance 2 graph of G 276 from the universal cover of G ⁢ ( Co 3 ) , in particular re-proving the simple connectedness of the latter geometry originally established in [A. Chermak, B. Oliver and S. Shpectorov, The linking systems of the Solomon 2-local finite groups are simply connected, Proc. Lond. Math. Soc. (3) 97 (2008), 1, 209–238]. A key step in our proof makes use of the simple connectedness of the Petersen-type geometry of the McLaughlin group, as shown in [B. Baumeister, A. A. Ivanov and D. V. Pasechnik, A characterization of the Petersen-type geometry of the McLaughlin group, Math. Proc. Cambridge Philos. Soc. 128 (2000), 1, 21–44].