|M.Sc Student||Ben David Nir|
|Subject||On Groups of Central Type|
|Department||Department of Mathematics||Supervisors||PROF. Eli Aljadeff|
|PROF. Shlomo Gelaki|
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(Γ, 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 : G → Aˇ. 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(Γ, 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.