M.Sc Thesis | |

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 **C**^{c}Γ is
a full matrix algebra over the field of complex numbers **C**, for some
2-cocycle *c*Î Z^{2}(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* → *A*ˇ. 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 *RES*_{A}^{Γ}
*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 H^{2}(Γ**,
C***).

Does there exist a
normal subgroup *A*Γ with |*A*|
= [Γ :* A*] such that *c* is trivial on *A* (i.e. [*c*]
Î *KER* *RES*_{A}^{Γ} )?

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