M.Sc Thesis
M.Sc Student Elimelekh Geva On Center-by-p-groups which Span Central Simple Algebras Department of Mathematics Full Professor Aljadeff Eli

Abstract

A finite dimensional k-central simple algebra A is called k-projective Schur if it is a homomorphic image of some twisted group algebra k α G where G is a finite group and α:G X G → k* is a 2-cocycle. Equivalently A is k-projective Schur if it is spanned over k by a subgroup Г≤A* of unit elements, which is finite modulo k*. We denote such an algebra by k (Г). We say that A is of p-type if it is k-projective and A=k(Г) where Г /k* is a p-group. A particular case of projective Schur algebras is when the twisted group algebra k αis k-central simple. In such a case, we say that the group G is of central type (non-classically). In 2005 Aljadeff, Haile and Natapov gave a short list of p-groups called Λ N p which consists exactly of those p-groups G for which there exists a field k and a 2-cocycle α such that k αis a division algebra. In this work we give a short list of p-groups called Λ G p . The groups on the list are a minimal set of groups for which any k-projective Schur algebra (char(k)=0) of p-type (not necessarily a division algebra) is Brauer equivalent to a k-projective Schur algebra k (Г) where Г /k*  is in Λ G p . Moreover every group G on the list Λ G p can be realized as a division algebra of the form k (Г)  where Г /k*  ≈ G and Г  has no proper  subgroup U such that  k (Г)=k(U). The list Λ G pN p   for odd primes however for p=2 the list Λ N 2  is a proper subset of Λ G p . Using this we give a bound for the index of k-projective Schur algebras of p-type. In addition we prove that any k-projective Schur algebra of p-type is Brauer equivalent to a tensor product of cyclic algebras.