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Summary

The most combinatorially interesting maximal subgroups of M24 are the stabilizers of an octad, a duum, a sextet and a trio. In this chapter we investigate the way in which the stabilizer of one of these objects acts on the others. This involves some basic but fascinating character theory; the approach given here is intended to be self-contained. For each of the four types of object we draw a graph in which each member is joined to members of the shortest orbit of its stabilizer. Thus in the octad graph we join two octads if they are disjoint; we join two dua if they cut one another 8.4/4.8; we join two sextets if the tetrads of one cut the tetrads of the other (22.04)6; and we join two trios if they have an octad in common. A diagram of each of these four graphs is included as is the way in which these graphs decompose under the action of one of the other stabilizers. Each of these graphs is, of course, preserved by M24.

Dedicated to the memory of Helmut Wielandt

In this paper a classification is obtained of all pairs G, H of finite primitive permutation groups such that G < H, the socles of G and H are different, and G and H share a common non-trivial orbital. An application is given to the study of automorphism groups of edge-transitive graphs.

2000 Mathematical Subject Classification: 20B15, 20B25.

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12 - Permutation Actions of M24

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PRIMITIVE PERMUTATION GROUPS WITH A COMMON SUBORBIT, AND EDGE-TRANSITIVE GRAPHS

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12 - Permutation Actions of M24

Summary

PRIMITIVE PERMUTATION GROUPS WITH A COMMON SUBORBIT, AND EDGE-TRANSITIVE GRAPHS