The O'Nan-Scott Theorem for Finite Primitive Permutation Groups, and Finite Representability

Abstract

The O'Nan-Scott Theorem classifies finite primitive permutation groups into one of five isomorphism classes. This theorem is very useful for answering questions about finite permutation groups since four out of the five isomorphism classes are well understood. The proof of this theorem currently relies upon the classification of the finite simple groups, as it requires a consequence of this classification, the Schreier Conjecture.

After reviewing some needed group theoretic concepts, I give a detailed proof of the O'Nan-Scott Theorem. I then examine how the techniques of this proof have been applied to an open problem which asks whether every finite lattice can be embedded as an interval into the subgroup lattice of a finite group.