#### Document Type

Dissertation

#### Date of Degree

Summer 2015

#### Degree Name

PhD (Doctor of Philosophy)

#### Degree In

Mathematics

#### First Advisor

Frauke Bleher

#### First Committee Member

Frauke Bleher

#### Second Committee Member

Victor Camillo

#### Third Committee Member

Miodrag Iovanov

#### Fourth Committee Member

Ryan Kinser

#### Fifth Committee Member

Muthu Krishnamurthy

#### Abstract

This thesis is on the representation theory of finite groups. Specifically, it is about finding connections between fusion and universal deformation rings.

Two elements of a subgroup *N* of a finite group Γ are said to be fused if they are conjugate in Γ, but not in *N*. The study of fusion arises in trying to relate the local structure of Γ (for example, its subgroups and their embeddings) to the global structure of Γ (for example, its normal subgroups, quotient groups, conjugacy classes). Fusion is also important to understand the representation theory of Γ (for example, through the formula for the induction of a character from *N* to Γ).

Universal deformation rings of irreducible mod *p* representations of Γcan be viewed as providing a universal generalization of the Brauer character theory of these mod *p* representations of Γ.

It is the aim of this thesis to connect fusion to this universal generalization by considering the case when Γ is an extension of a finite group *G* of order prime to *p* by an elementary abelian *p*-group *N* of rank 2. We obtain a complete answer in the case when *G* is a dihedral group, and we also consider the case when *G* is abelian. On the way, we compute for many absolutely irreducible **F**_{p}Γ-modules *V*, the cohomology groups H^{2}(Γ,Hom_{Fp}(V,V) for *i* = 1, 2, and also the universal deformation rings *R*(Γ,V).

#### Public Abstract

This thesis discusses to what extent the universal deformation rings of representations can be used to detect fusion in group theory. Groups, rings, and representations are three important areas of study in Abstract Algebra. This thesis connects group-theoretic phenomena to representations of certain infinite families of groups by associating a ring, the universal deformation ring, to special representations of each group. In the cases considered, the universal deformation rings of these representations will typically be the same. However, the knowledge of when the universal deformation rings are different can be used to detect the fusion of a certain normal subgroup in each larger group under consideration. In this sense, universal deformation rings can “see” fusion. This thesis develops the machinery necessary to investigate this connection in general. It then analyzes the connection completely for some infinite classes of groups.

#### Keywords

publicabstract, Group theory, Homological algebra, Representation theory, Universal deformation rings

#### Pages

v, 80 pages

#### Bibliography

Includes bibliographical references (pages 79-80).

#### Copyright

Copyright 2015 David Christopher Meyer