CEU Electronic Theses and Dissertations, 2009
| Author | Halasi, Zoltan |
|---|---|
| Title | On the representations of solvable linear groups |
| Summary | We discuss three topic concerning the representations of solvable linear groups. First, we give a positive answer to a question of I. M. Isaacs about the characters of finite algebra groups. To answer Isaacs' question we prove a new identity for the commutators in finite algebra groups. To confirm this identity we use Lie theoretic methods. Then we generalise Isaacs' question to the unit group of a DN-algebra by using ordinary character theory. Next, we examine a natural generalisation of a conjecture of G. Higman about the number of conjugacy classes in the group of upper unitriangular matrices. We prove that the analogue of Higman's conjecture does not hold to the so-called partition subgroups by using linear algebra and a few algebraic geometry. Finally, we give a partly constructive proof to the widely asked conjecture that if G is a solvable linear group acting on V such that (|G|, |V|) = 1, then there exist x, y in V such that only the identity element of G fixes both x and y. To find such a pair of vectors we use tools from the theory of permutation groups, some linear algebra, some representation theory of finite groups and a nice description of maximal solvable primitive linear groups. |
| Supervisor | Palfy, Peter Pal |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2009/tphhaz01.pdf |
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