Some Chamber Systems Belonging to Sporadic Simple Groups

The aim of this paper is to study a certain type of chamber system. For notation and definitions concerning chamber systems see [22]. For the convenience of the reader, we recall two definitions which are of importance in this paper. Let % = (% ($,•),-€/) be a chamber system over / and c € %. By Ay(c) with / c /, we denote the element in £%y containing c (Ay(c) is a chamber system over / ) . Let G be a subgroup of the automorphism group of <# which acts transitively on the chambers. We denote by Gy the stabilizer of A,_y(c), and by Kj the kernel of the action on A7_y(c). For / el, we use Pt to denote GA.(c), and G, to denote G/_{l|. In the sequel we will consider only connected chamber systems <#. This is the same as saying G = {Pt\ i el). We assume the following hypotheses. Let % be a connected chamber system over / with chamber-transitive automorphism group G, such that B = Gc is finite for c a chamber. For i e I, we have G±.{c)lKA.{c) = S3 or S5. For / c /, |7 J\ = 2, we have (i) GJ/KJ^L3(2), A6> 26, 3-A6, 3 • 56, l/3(3), G2(2), M12, Aut(MI2), G2(3), G2(3)-2, 3-G2(3), 3-G2(3)-2, (ii) Gj/Kj = Aut(J2), LyS, HiS, Aut(HiS), Ru, M22, Aut(M22), 3 • M22, 3 • Aut(M22), U4(2) • 2, or (iii) GjlKj = 53 x S3) S5 x S5, (Z3 x Z3) • 2, (,45 x /l5) • 2, 53 x 55, (A5 x Z3) • 2. REMARK. Let / -J = {*, y}. Then Gy = {/>.,/>.). In Case (i) we have P;IG&.{c) = (C) = S3. If GJ/KJ is a Lie group or a covering of a Lie group, then PJKj and y are just the minimal parabolics. If GJ/KJ involves Af12, then there are exactly two maximal subgroups containing a Sylow 2-subgroup. These are PjlKj and Pj/Kj. In G2(3) there are five subgroups X containing a Sylow 2-subgroup S such that X/O2(X) = 53. Four of them are in C(Z(5)). Let Y be the one which is not in C(Z(5)). Thus there is exactly one XcC(Z(5)) , with X/O2(X) = S3 such that (X, Y) = G2(3). So if GJ/KJ involves G2(3), we have that PJKj, PJ/KJ correspond to X and Y. In Case (ii) we have /^/GA.(c) = 53, ^/GA.(c) = 55. The chamber system has been described in [13]. In Case (iii) we have Gy = P/Pj. So in all cases the chamber system A7_y(c) is uniquely determined by the structure of GJ/KJ.