An explicit formula for the action of a finite group on a commutative ring

Abstract

Let G be a finite group, k a commutative ring upon which G acts. For every subgroup H of G , the trace (or norm) map tr H : k → k H is defined. tr H is onto if and only if there exists an element x H such that tr H ( x H ) = 1 . We will show that the existence of x P for every subgroup P of prime order determines the existence of x G by exhibiting an explicit formula for x G in terms of the x P , where P varies over prime order subgroups. Since tr P is onto if and only if tr g P g − 1 is, where g ∈ G is an arbitrary element, we need to take only one P from each conjugacy class. We will also show why a formula with less factors does not exist, and show that the existence or non-existence of some of the x P ’s (where we consider only one P from each conjugacy class) does not affect the existence or non-existence of the others.