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

TL;DR

This paper derives an explicit formula for how a finite group acts on a commutative ring, enabling the expression of certain elements in terms of subgroup elements with specific trace properties.

Contribution

It introduces an induction formula that relates elements with trace one across the entire group to those in prime order subgroups, providing a new computational tool.

Findings

Derived an explicit induction formula for group actions on rings

Connected elements with trace one across the group to prime order subgroups

Enhanced understanding of group actions in algebraic structures

Abstract

Let G be a group which acts on a commutative ring k. We exhibit an induction formula which expresses an element x_G with tr_G(x_G)=1 by elements x_P with tr_P(x_P)=1, where P varies over prime order subgroups of P.