Oldenburg, University, Diss., 2001. / Dateiformat: zip, Dateien in unterschiedlichen Formaten.

A Schur ring is a subring of the group algebra with a basis that is formed by a partition of the group. These subrings were initially used to study finite permutation groups, and classifications of Schur rings over various finite groups have been studied. Here we investigate Schur rings over various infinite groups, including free groups. We classify Schur rings over the infinite cyclic group.

We study finite groups $G$ having a nontrivial subgroup $H$ and $D \subset G \setminus H$ such that (i) the multiset $\{ xy^{-1}:x,y \in D\}$ has every element that is not in $H$ occur the same number of times (such a $D$ is called a {\it relative difference set}); (ii) $G=D\cup D^{(-1)} \cup H$; (iii) $D \cap D^{(-1)} =\emptyset$. We show that $|H|=2$, that $H$ has to be normal, and that $G$ is a group with a single involution. We also show that $G$ cannot be abelian. We give examples of such groups, including certain dicyclic groups, by using results of Schmidt and Ito. We describe an infinite family of dicyclic groups with these relative difference sets, and classify which groups of order up to $72$ contain them. We also define a relative difference set in dicyclic groups having additional symmetries, and completely classify when these exist in generalized quaternion groups. We make connections to Schur rings and prove additional results.

In this thesis, we will explore the structure of Terwilliger algebras over several different types of finite groups. We will begin by discussing what a Schur ring is, as well as providing many different results and examples of them. Following our discussion on Schur rings, we will move onto discussing association schemes as well as their properties. In particular, we will show every Schur ring gives rise to an association scheme. We will then define a Terwilliger algebra for any finite set, as well as discuss basic properties that hold for all Terwilliger algebras. After specializing to the case of Terwilliger algebras resulting from the orbits of a group, we will explore bounds of the dimension of such a Terwilliger algebra. We will also discuss the Wedderburn decomposition of a Terwilliger algebra resulting from the conjugacy classes of a group for any finite abelian group and any dihedral group.

In this dissertation, we explore the nature of Schur rings over finite cyclic groups, both algebraically and combinatorially. We provide a survey of many fundamental properties and constructions of Schur rings over arbitrary finite groups. After specializing to the case of cyclic groups, we provide an extensive treatment of the idempotents of Schur rings and a description for the complete set of primitive idempotents. We also use Galois theory to provide a classification theorem of Schur rings over cyclic groups similar to a theorem of Leung and Man and use this classification to provide a formula for the number of Schur rings over cyclic p-groups.