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 FpΓ-modules V, the cohomology groups H2(Γ,HomFp(V,V) for i = 1, 2, and also the universal deformation rings R(Γ,V).
| Identifer | oai:union.ndltd.org:uiowa.edu/oai:ir.uiowa.edu:etd-5940 |
| Date | 01 July 2015 |
| Creators | Meyer, David Christopher |
| Contributors | Bleher, Frauke, 1968- |
| Publisher | University of Iowa |
| Source Sets | University of Iowa |
| Language | English |
| Detected Language | English |
| Type | dissertation |
| Format | application/pdf |
| Source | Theses and Dissertations |
| Rights | Copyright 2015 David Christopher Meyer |
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