On the Minimal Parabolic System Related to the Monster Simple Group

Almost all of the 26 sporadic finite simple groups possess systems of subgroups which, in several ways, resemble the parabolic systems of the finite simple groups of Lie type. Many of these systems are catalogued in [RSt, RSm]. Those in [RSt] mirror more closely the minimal parabolic systems of a finite simple group of Lie type. In this paper the term minimal parabolic system will be used in a broader sense, to be explained in a moment. Our interest here is in a very particular kind of (generalized) minimal parabolic system, an example of which occurs in the Monster simple group, and the object of this work is to derive ``local'' information about such a minimal parabolic system. In order to state these results we need to introduce some definitions and notation.