representation ring vs burnside ring

Let $G$ be a finite group and let $k$ be any field. If $X$ is a $G$-set, then we may consider the vector space $V_k(X)$ over $k$ which has $X$ as a basis. In this manner $V_k(X)$ becomes a representation of $G$ via action induced from $X$ and linearly extended to $V_k(X)$. It can be shown that $V_k(X)$ only depends on the isomorphism class of $X$, so we have a well-defined mapping:

$$[X]\mapsto [V_k(X)]$$

which can be easily extended to the function

$$\beta :\mathrm{\Omega }(G)\to R_k(G);$$

$$\beta ([X])=[V_k(X)]$$

where on the left side we have the Burnside ring and on the right side the representation ring. It can be shown, that $\beta$ is actually a ring homomorphism, but in most cases it neither injective nor surjective. But the following theorem due to Segal gives us some properties of $\beta$:

Theorem (Segal). Let $\beta :\mathrm{\Omega }(G)\to R_{\mathbb{Q}}(G)$ be defined as above with rationals as the underlying field. If $G$ is a $p$-group for some prime number $p$, then $\beta$ is surjective. Furthermore $\beta$ is an isomorphism if and only if $G$ is cyclic.

Title	representation ring vs burnside ring
Canonical name	RepresentationRingVsBurnsideRing
Date of creation	2013-03-22 19:19:05
Last modified on	2013-03-22 19:19:05
Owner	joking (16130)
Last modified by	joking (16130)
Numerical id	4
Author	joking (16130)
Entry type	Theorem
Classification	msc 20C99