טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentNir Ben David
SubjectOn Groups of Central Type
DepartmentDepartment of Mathematics
Supervisors Full Professor Aljadeff Eli
Full Professor Gelaki Shlomo


Abstract

Let Γ be a finite group. We say that Γ is of central type (nonclassically) if the twisted group algebra CcΓ is a full matrix algebra over the field of complex numbers C, for some 2-cocycle cÎ Z2, C*).

In this case c is said to be nondegenerate. An important family of central type groups was constructed  by Etingof and Gelaki. Let G be a group of order n which acts on an abelian group A of order n.

Suppose that there is a bijective 1-cocycle π : GAˇ. Then the semidirect product Γ = A G is  a group of central type. Moreover, we have an explicit formula for the nondegenerate 2-cocycle.

The first main result of this work is a generalization of this construction to an arbitrary extension  1 → A → Γ → G → 1. The second result is to show that any nilpotent central type group Γ with nondegenerate 2-cocycle c has a subgroup A such that |A| = [Γ : A] and RESAΓ c = 1. Moreover,

if A is normal then A is abelian.


Question: Let Γ be a nilpotent group of central type with a nondegenerate 2-cocycle [c] Î H2, C*).

Does there exist a normal subgroup AΓ with |A| = [Γ : A] such that c is trivial on A (i.e. [c] Î KER RESAΓ )?

If the answer is positive then any nilpotent group of central type can be constructed by the above method.