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 trH:k→kH is defined. trH is onto if and only if there exists an element xH such that trH(xH)=1. 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 trP is onto if and only if trgPg−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 xP's (where we consider only one P from each conjugacy class) does not affect the existence or non-existence of the others.