Generalized group characters and complex oriented cohomology theories
Michael J. Hopkins; Nicholas J. Kuhn; Douglas C. Ravenel. 
J. Amer. Math. Soc. 13 (2000), 553-594.

                                
Abstract: Let $BG$ be the classifying space of a finite group $G$. 
Given a multiplicative cohomology theory $E^*$, the assignment 
\[ G \mapsto E^*(BG) \]
is a functor from groups to rings, endowed with induction (transfer) maps. 
In this paper we investigate these functors for {\em complex oriented\/} 
cohomology theories $E^*$, using the theory of complex representations of 
finite groups as a model for what one would like to know. 

An analogue of Artin's Theorem is proved for all complex oriented $E^*$: 
the abelian subgroups of $G$ serve as a detecting family for $E^*(BG)$, 
modulo torsion dividing the order of $G$. 

When $E^*$ is a complete local ring, with residue field of characteristic
$p$ and associated formal group of height $n$, we construct a character 
ring of class functions that computes $\frac{1}{p}E^*(BG)$. The domain of 
the characters is $G_{n,p}$, the set of $n$-tuples of elements in $G$ each 
of which has order a power of $p$. A formula for induction is also found. 
The ideas we use are related to the Lubin-Tate theory of formal groups. The
construction applies to many cohomology theories of current interest: 
completed versions of elliptic cohomology, $E_n^*$-theory, etc. 

The $n$th Morava K-theory Euler characteristic for $BG$ is computed to be 
the number of $G$-orbits in $G_{n,p}$. For various groups $G$, including 
all symmetric groups, we prove that $K(n)^*(BG)$ is concentrated in even 
degrees. 

Our results about $E^*(BG)$ extend to theorems about $E^*(EG \times_G X)$, 
where $X$ is a finite $G$-CW complex.