The O'Nan–Scott Theorem

Abstract

Introduction We have already seen two reductions for a finite permutation group: (a) An intransitive group is a subcartesian product of its transitive constituents. (b) A transitive but imprimitive group is contained in the iterated wreath product of its primitive components. These enable many questions about arbitrary groups to be reduced to the case of a primitive group and hopefully solved there. In this section, we make one further reduction, and then attempt to describe the ‘basic’ building blocks we have reached. The socle of a finite group G is the product of the minimal normal subgroups of G . (The original meaning of the word is ‘the base on which a statue stands’.) The socle is a normal subgroup whose structure can be described: it is a direct product of finite simple groups (but, in general, not all of these simple groups are isomorphic). We will see that, for a primitive group G , either the socle is a product of isomorphic finite simple groups in a known permutation action, or G itself is almost simple. In many practical problems, we can then appeal to the Classification of Finite Simple Groups (CFSG) to reach a conclusion in the last case. A version of this theorem (though without much of the detail) which is sufficient for some of the applications appears in Jordan's Traite des Substitutions . But since the use of the theorem depends on knowledge of the finite simple groups which was not available for 120 years after Jordan, his result was forgotten.