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

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 image is defined. image is onto if and only if there exists an element xH such that image. We will show that the existence of xP for every subgroup P of prime order determines the existence of xG by exhibiting an explicit formula for xG in terms of the xP, where P varies over prime order subgroups. Since image is onto if and only if image is, where gset membership, variantG 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 xP’s (where we consider only one P from each conjugacy class) does not affect the existence or non-existence of the others.