$K$-homology of $BG$ Let $G$ be a finite group.  Atiyah proved that the $K$-cohomology of $BG$ vanishes in odd degrees and in even degrees is the completion of the representation ring of $G$ at the augmentation ideal.
What is the $K$-homology of $BG$?  I mean the stable homotopy groups of the complex $K$-theory spectrum smash $BG$.
 A: For finite-type torsion spectra $X$ there is a natural isomorphism 
$$ KU_{n-1}(X) \simeq \text{Hom}_c(KU^n(X),\mathbb{Q}/\mathbb{Z}). $$
Here $\text{Hom}_c$ denotes the group of homomorphisms that are continuous with respect to the skeletal topology on $KU^*(X)$ and the discrete topology on $\mathbb{Q}/\mathbb{Z}$.  The isomorphism can be obtained in a fairly straightforward way using $X\wedge S\mathbb{Q}=0$ (so $X=\Sigma^{-1}X\wedge S(\mathbb{Q}/\mathbb{Z})$) and standard exactness arguments. 
If $G$ is a finite group then we can take $X$ to be $\Sigma^\infty BG$ (without disjoint basepoint) to get reduced $K$-homology groups $KU_{2n}(BG)=0$ and
$$ KU_{2n-1}(BG) = \text{Hom}_c(\widehat{J}(G),\mathbb{Q}/\mathbb{Z}). $$
Here $J(G)$ is the augmentation ideal in the representation ring, and $\widehat{J}(G)$ is the completion of $J(G)$ with respect to itself.  If $G$ is a $p$-group then we just have $\widehat{J}(G)=J(G)\otimes\mathbb{Z}_p$.
The above answer is of course the same as you get via local cohomology, but this argument is more elementary.
A: In "$K$-homology of universal spaces and local cohomology of the representation ring" Greenlees proves that 
$$
K_i(BG_+) \simeq H_J^i(R(G))
$$
for $i=0,1$, where $J$ is the augmentation ideal and $H_J^i(R(G))$ refers to local cohomology. In fact Greenlees proves something more general using equivariant homotopy theory (for which the above is a special case), and this more general object is also studied in Chapter 18 of Greenlees and May's "Generalized Tate Cohomology"
Equivalently, one can use Anderson's universal coefficient theorem for $K$-theory (see  here). Anderson shows that if $G$ is even a compact connected Lie group that $KU_1(BG) = 0$ and $$KU_0BG = \operatorname{lim}_n\operatorname{Hom}(R(G)/J^n,\mathbb{Z}).$$
There is a discussion of the relationship between the two results in Remark 2.8 of "Topological $K$-(co)homology of classifying spaces of discrete groups", by Joachim and  Lück (which also contains more results along these lines). 
