Thesis (Ph. D.)--University of Texas at Austin, 2006. / Vita. Includes bibliographical references.

Not available / text

This thesis focuses on the subject of varieties for modules for elementary abelian p-groups. Given a homogeneous polynomial over an algebraically closed field of char- acteristic 2 we will give constructions for modules of small dimension having that polynomial as variety. This is similar to an earlier construction given by Jon Carlson but our modules will in general be of considerably smaller dimension. We also investigate the connection between the variety of a module and its Loewy length. We show that working over an algebraically closed field of characteristic 2 with modules of Loewy length 2 allows us to find modules with any hypersurface as their variety. On the other hand we also demonstrate that in odd characteristic p, with modules of Loewy length p, the only possible varieties are finite unions of linear hypersurfaces.

The thesis centres around two problems in the enumeration of p-groups. Define f_φ(p^m) to be the number of (isomorphism classes of) groups of order p^m in an isoclinism class φ. We give bounds for this function as φ is fixed and m varies and as m is fixed and φ varies. In the course of obtaining these bounds, we prove the following result. We say a group is reduced if it has no non-trivial abelian direct factors. Then the rank of the centre Z(P) and the rank of the derived factor group P|P' of a reduced p-group P are bounded in terms of the orders of P|Z(P)P' and P'∩Z(P). A long standing conjecture of Charles C. Sims states that the number of groups of order p^m is<br/> p^{²andfrasl;₂₇m³+O(m²)}. (1) We show that the number of groups of nilpotency class at most 3 and order p^m satisfies (1). We prove a similar result concerning the number of graded Lie rings of order p^m generated by their first grading.

No description available.

No description available.

No description available.

In 1935, Philip Hall developed a formula for finding the number of ways of generating the group of symmetries of the icosahedron from a given number of its elements. In doing so, he defined a generalized Eulerian function. This thesis uses Hall's generalized Eulerian function to calculate generalized Eulerian functions for specific groups, namely: cyclic groups, dihedral groups, and p- groups.

The Atiyah Conjecture originates from a paper written 40 years ago by Sir Michael Atiyah, a famous mathematician and Fields medalist. Since publication of the paper, mathematicians have been working to solve many questions related to the conjecture, but it is still open. The conjecture is about certain topological invariants attached to a group 𝐺. There are examples showing that the conjecture does not hold in general. These examples involve something like the lamplighter group (the wreath product ℤ/2ℤ ≀ ℤ). We are interested in looking at examples where this is not the case. We are interested in the specific case where 𝐺 is a finitely generated group in which the Prüfer group can be embedded as the center. The Prüfer group is a 𝑝-group for some prime 𝑝 and its finite subgroups have unbounded order, in particular the finite subgroups of G will have unbounded order. To understand whether any form of the Atiyah conjecture is true for 𝐺, it will first help to determine whether the group ring 𝑘𝐺 of the group 𝐺 has a classical ring of quotients for some field 𝑘. To determine this we will need to know the zero divisors for the group ring 𝑘𝐺. Our investigations will be divided into two cases, namely when the characteristic of the field 𝑘 is the same as the prime p for the Prüfer group and when it is different. / Master of Science

Representation theory is concerned with the ways of writing elements of abstract algebraic structures as linear transformations of vector spaces. Typical structures amenable to representation theory are groups, associative algebras and Lie algebras. In this thesis we study linear representations of finite groups. The study focuses on character theory and how character theory can be used to extract information from a group. Prior to that, concepts needed to treat character theory, and some of their ramifications, are investigated. The study is based on existing literature, with excessive use of examples to illuminate important aspects. An example treating a class of p-groups is also discussed.