# K-theory of compact Lie groups

The complex K-theory of a compact connected Lie group $G$ is computed by Hodgkin in the case that $G$ has torsion-free fundamental group. The result is that $K^*(G)$ is an exterior algebra in the fundamental representations of $G$. Note that every finite dimensional representation gives a canonical $K^1$-class.

What happens if $\pi_1(G)$ has torsion? Are there any general results analogous to Hodgkin's theorem? I am specifically interested in the case of the projective unitary group $PU(n) = U(n)/center$. In the case $n = 2$ this amounts to $SO(3)$, and identification with $\mathbb{R}P^3$ shows that $K^0$ has a torsion subgroup $\mathbb{Z}/2\mathbb{Z}$ in this case. I would expect to find a torsion subgroup $\mathbb{Z}/n\mathbb{Z}$ inside $K^0$ in general - is this true?

-