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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?

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