M.Sc Thesis
M.Sc Student Ehud Meir Ben Efraim An Explicit Formula for the Action of a Finite Group on a Commutative Ring Department of Mathematics Full Professor Sonn Jack Full Professor Aljadeff Eli

Abstract

The main object of this work is to find an explicit formula for the action of a finite group on a commutative ring. Let G be a finite group which acts upon a commutative ring k. We can define the trace map trG:k à kG, and ask ourselves when this map is onto. This question is of interest for us, because it turns out that trG is onto if and only if k is projective as a module over the twisted group ring ktG. It turns out, since trG is kG linear, that trG is onto if and only if there exists an element xG in k, such that trG(xG)=1. It is known that if trG is onto, then trH is onto, for every subgroup H of G. The main focus of this work will be to deduce the other direction: Suppose we know that for some of the subgroups H of G, trH is onto. When can we deduce that trG is onto as well? Equivalently, we may ask when the existences of xH, for some of the subgroups H of G determine the existence of xG. It is known, for example, that if H is a subgroup of G such that there exists an element xH, and for every subgroup N of G that intersects H trivially, there exists an element xN, then there exists an element xG. Moreover, there exists a polynomial formula which expresses xG in terms of the elements xH and xN. In this work we will find a formula for xG in terms of the elements xP, where P varies over the subgroup of G of prime order. We will also give several examples for using the two formulas mentioned above. In addition, we will discuss the case where the ring k is not commutative. It is known that in this case, the existence of xE, for every elementary abelian subgroup E of G, determines the existence of xG. However, there is not a known formula for xG in terms of the xE's. We shall give such a formula in the case where G is the cyclic group of order p2, for p a prime number. We will also show that the existence (or non-existence) of some of the xP's, does not determine the existence (or non-existence) of the others.