#### DOI

10.17077/etd.a1y45gp0

#### Document Type

Dissertation

#### Date of Degree

Summer 2015

#### Degree Name

PhD (Doctor of Philosophy)

#### Degree In

Mathematics

#### First Advisor

Frauke Bleher

#### First Committee Member

Victor Camillo

#### Second Committee Member

Miodrag Iovanov

#### Third Committee Member

Ryan Kinser

#### Fourth Committee Member

Philip Kutzko

#### Abstract

The main objective of deformation theory is to study the behavior of mathematical objects, such as modules or group representations, under perturbations. This theory is useful in both pure and applied mathematics and has led to the solution of many long-standing problems. For example, in number theory, universal deformation rings of Galois representations played an important role in the proof of Fermat’s Last Theorem by Wiles and Taylor.

In this thesis, we consider the case when *SD _{n}* is a semidihedral 2-group of order 2

^{n+1}for n ≥ 3 and

*k*is an algebraically closed field of characteristic 2. The indecomposable

*kSD*-modules have been completely described by Bondarenko and Drozd, and Crawley-Boevey. We concentrate on so-called endo-trivial

_{n}*kSD*-modules, which possess a well-defined universal deformation ring by work of Bleher and Chinburg. Using the classification of Carlson and Thevenaz of all endo-trivial

_{n}*kSD*-modules, we show that the universal deformation ring of every endo-trivial

_{n}*kSD*-module is isomorphic to the group ring W [ℤ/2 x ℤ/2], where W = W (k) is the ring of infinite Witt vectors over k.

_{n}#### Public Abstract

The main objective of deformation theory is to study the behavior of mathematical objects, such as group representations, under perturbations. This theory is useful in both pure and applied mathematics and has led to the solution of many long-standing problems. For example, in number theory, universal deformation rings of Galois representations played an important role in the proof of Fermat's Last Theorem by Wiles and Taylor.

In this thesis, we consider the case of representations of semidihedral 2-groups in characteristic 2. The characteristic 2 representation theory of semidihedral 2-groups is well understood by work of Bondarenko and Drozd, and Crawley-Boevey. Moreover, the so-called endo-trivial characteristic 2 representations of these groups have been classified by Carlson and Thévenaz. We use this classification to determine the universal deformation ring of every endo-trivial characteristic 2 representation of every semidihedral 2-group of order 16 or higher.

#### Keywords

publicabstract

#### Pages

vii, 81 pages

#### Bibliography

Includes bibliographical references (pages 80-81).

#### Copyright

Copyright 2015 Roberto Carlos Soto

#### Recommended Citation

Soto, Roberto Carlos. "Universal deformation rings and semidihedral 2-groups." PhD (Doctor of Philosophy) thesis, University of Iowa, 2015.

https://doi.org/10.17077/etd.a1y45gp0