# functoriality of the Burnside ring

We wish to show how the Burnside ring $\mathrm{\Omega}$ 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 $\mathrm{\Omega}$ acts on objects of the category of finite groups. Assume that $f:G\to H$ is a group homomorphism^{}. Furthermore let $X$ be a $H$-set. Then $X$ can be naturally equiped with a $G$-set structure^{} via function:

$$(g,x)\u27fcf(g)x.$$ |

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

Therefore a group homomorphism $f:G\to H$ induces a ring homomorphism^{}

$$\mathrm{\Omega}(f):\mathrm{\Omega}(H)\to \mathrm{\Omega}(G)$$ |

such that

$$\mathrm{\Omega}(f)([X]-[Y])=[{X}_{f}]-[{Y}_{f}].$$ |

One can easily check that this turns $\mathrm{\Omega}$ 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 | Derivation^{} |

Classification | msc 16S99 |