An explicit formula for the action of a finite group on a commutative ring. (Q2371811)

Let \(G\) be a finite group, \(K\) be a commutative ring upon which \(G\) acts. For any subgroup \(H\) of \(G\), the trace map \(\text{tr}_H\colon K\to K^H\) is defined. \(\text{tr}_H\) is onto if and only if there exists an element \(x_H\) such that \(\text{tr}_H(x_H)=1\). In this paper, the author shows 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 \(x_P\), where \(P\) varies over prime order subgroups.