A Study of Rank Four Permutation Groups

Citation

Renfrow, James Thomas (1969) A Study of Rank Four Permutation Groups. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/WPHQ-7W89. https://resolver.caltech.edu/CaltechETD:etd-09272002-154545

Abstract

NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document. In this thesis we study rank 4 permutation groups. A rank 4 group is a finite transitive permutation group acting on a set [Omega] such that the subgroup fixing a letter breaks up [Omega] into 4 orbits. The main tool employed in examining rank 4 groups is the use of intersection matrices, an idea introduced by Donald Higman. Intersection matrices are used to obtain relations between the lengths of the four orbits associated with a rank 4 representation and the degrees of the irreducible characters in the permutation character of the representation. It is shown that two orbits of the representation are paired if and only if two of the characters are complex conjugates of one another. All the maximal primitive rank 4 groups are determined. Techniques are developed, using intersection matrices, to find all rank 4 representations of known finite groups. Group theoretic results about possible rank 4 groups are derived from the intersection matrices which would have to correspond to the rank 4 representation.

Thesis Files