In this paper we study the orbits of the Mathieu group M24 on sets of n points, 1 6, we know that for n > 6 there will be more than one type of n's. If there are j different types of sets of n distinct points, they are denoted by n', n, . . ., ni. G(n') denotes the setwise stabilizer of ni in M24 and G[ni] denotes the pointwise stabilizer of ni in M24. We will adopt one more notation, A,(ni). This denotes an orbit of length s of G(n') on Q ni. Presented to the Society, August 30, 1968 under the title The complete list of maximal subgroups of Mathieu group of degree 24; received by the editors August 3, 1970. AMS 1969 subject classifications. Primary 2020, 2029.
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