On Zn ⋊ Z2-Hopf-Galois structures and unit group of some group algebras

Abstract

This thesis is divided in two parts. The first part talks about Hopf-Galois structures on groups of the form Zn⋊φZ2. Let K/F be a finite Galois extension of fields with Gal(K/F) = Γ. We enumerate the Hopf-Galois structures with Galois group Γ of type G, where Γ, G are groups of the form Zn ⋊φ Z2 when n is odd with radical of n being a Burnside number. These findings have applications in the study of solutions to the Yang-Baxter equations and also give application in the field of Galois module theory. The second part entails unit groups of some finite semisimple group algebra. This is further divided into two subsections. Firstly we provide the structure of the unit group of Fpk (SL(3, 2)), where p ≥ 11 is a prime and SL(3, 2) denotes the 3×3 invertible matrices over F2. Secondly we give the structure of the unit group of Fpk Sn, where p > n is a prime and Sn denotes the symmetric group on n letters. This provide the complete characterization of the unit group of the group algebra Fpk A6 for p ≥ 7, where A6 is the alternating group on 6 letters.