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 → k H 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.