# functoriality of the Burnside ring

We wish to show how the Burnside ring $\Omega$ can be turned into a contravariant functor from the category of finite groups into the category of commutative, unital rings.

Let $G$ and $H$ be finite groups. We already know how $\Omega$ acts on objects of the category of finite groups. Assume that $f:G\rightarrow H$ is a group homomorphism. Furthermore let $X$ be a $H$-set. Then $X$ can be naturally equiped with a $G$-set structure via function:

 $(g,x)\longmapsto f(g)x.$

The set $X$ equiped with this group action will be denoted by $X_{f}$.

Therefore a group homomorphism $f:G\rightarrow H$ induces a ring homomorphism

 $\Omega(f):\Omega(H)\rightarrow\Omega(G)$

such that

 $\Omega(f)([X]-[Y])=[X_{f}]-[Y_{f}].$

One can easily check that this turns $\Omega$ into a contravariant functor.

Title functoriality of the Burnside ring FunctorialityOfTheBurnsideRing 2013-03-22 18:08:06 2013-03-22 18:08:06 joking (16130) joking (16130) 5 joking (16130) Derivation msc 16S99