functoriality of the Burnside ring

We wish to show how the Burnside ring Ω can be turned into a contravariant functorMathworldPlanetmath from the categoryMathworldPlanetmath of finite groupsMathworldPlanetmath into the category of commutativePlanetmathPlanetmathPlanetmath, unital rings.

Let G and H be finite groups. We already know how Ω acts on objects of the category of finite groups. Assume that f:GH is a group homomorphismMathworldPlanetmath. Furthermore let X be a H-set. Then X can be naturally equiped with a G-set structureMathworldPlanetmath via function:


The set X equiped with this group actionMathworldPlanetmath will be denoted by Xf.

Therefore a group homomorphism f:GH induces a ring homomorphismMathworldPlanetmath


such that


One can easily check that this turns Ω into a contravariant functor.

Title functoriality of the Burnside ring
Canonical name FunctorialityOfTheBurnsideRing
Date of creation 2013-03-22 18:08:06
Last modified on 2013-03-22 18:08:06
Owner joking (16130)
Last modified by joking (16130)
Numerical id 5
Author joking (16130)
Entry type DerivationPlanetmathPlanetmath
Classification msc 16S99