# FiniteGroupAction -- the class of all finite group actions

## Description

This class is provided by the package InvariantRing.

FiniteGroupAction is the class of all finite matrix group actions on polynomial rings for the purpose of computing invariants. It is created using finiteAction. Note that diagonal actions of finite abelian groups can be created with the class DiagonalAction which implements more efficient methods for computing invariants.

## Functions and methods returning an object of class FiniteGroupAction :

• "finiteAction(List,PolynomialRing)" -- see finiteAction -- the group action generated by a list of matrices
• "finiteAction(Matrix,PolynomialRing)" -- see finiteAction -- the group action generated by a list of matrices

## Methods that use an object of class FiniteGroupAction :

• generators(FiniteGroupAction) -- generators of a finite group
• "group(FiniteGroupAction)" -- see group -- list all elements of the group of a finite group action
• "hironakaDecomposition(FiniteGroupAction)" -- see hironakaDecomposition -- calculates a Hironaka decomposition for the invariant ring of a finite group
• invariants(FiniteGroupAction) -- computes the generating invariants of a group action
• "invariants(FiniteGroupAction,List)" -- see invariants(FiniteGroupAction,ZZ) -- basis for graded component of invariant ring
• invariants(FiniteGroupAction,ZZ) -- basis for graded component of invariant ring
• "isAbelian(FiniteGroupAction)" -- see isAbelian -- check whether a finite matrix group is Abelian
• "isInvariant(RingElement,FiniteGroupAction)" -- see isInvariant -- check whether a polynomial is invariant under a group action
• "molienSeries(FiniteGroupAction)" -- see molienSeries -- computes the Molien (Hilbert) series of the invariant ring of a finite group
• "net(FiniteGroupAction)" -- see net(RingOfInvariants) -- format for printing, as a net
• numgens(FiniteGroupAction) -- number of generators of a finite group
• "primaryInvariants(FiniteGroupAction)" -- see primaryInvariants -- computes a list of primary invariants for the invariant ring of a finite group
• relations(FiniteGroupAction) -- relations of a finite group
• "reynoldsOperator(RingElement,FiniteGroupAction)" -- see reynoldsOperator -- the image of a polynomial under the Reynolds operator
• "schreierGraph(FiniteGroupAction)" -- see schreierGraph -- Schreier graph of a finite group
• "secondaryInvariants(List,FiniteGroupAction)" -- see secondaryInvariants -- computes secondary invariants for the invariant ring of a finite group
• "words(FiniteGroupAction)" -- see words -- associate a word in the generators of a group to each element

## For the programmer

The object FiniteGroupAction is a type, with ancestor classes GroupAction < HashTable < Thing.