Finite primitive permutation groups of rank 4

Abstract

In this thesis we classify finite primitive permutation groups of rank 4. According to the 0' Nan-Scott theorem, a finite primitive permutation group is an affine group, an almost simple group, or has either simple diagonal action, product action or twisted wreath action. In Chapter 1 we completely determine the primitive rank 4 permutation groups with one of the last three types of actions up to permutation equivalence. In Chapter 2 we use Aschbacher's subgroup structure theorem for the finite classical groups to reduce the classification of affine primitive rank 4 permutation groups G of degree p*^ (p prime) to the case where a point stabilizer G in G satisfies soc(G/Z(G ))=L for some ^ 0 0 0 non-abelian simple group L. In Chapter 3 we classify all such groups G with L a simple group of Lie type over a finite field of characteristic p. Finally, in Chapter 4 we determine all the faithful primitive rank 4 permutation representations of the finite linear groups up to permutation equivalence.