Letting $C_n$ be the cyclic group on $n$ elements we know through the use of the Atiyah-Segal completion theorem that $$ K^*(BG) = \pi_*(KU)[[t]]/((t+1)^n -1) $$ where $\pi_*(KU)=\mathbb{Z}[u^{\pm 1}]$ with $u$ being the Bott periodicity element in $\pi_2(KU)$. I was wondering if it was possible to do something similar to caluclate real $K$-theory $KO^*(BC_n)$ for cyclic groups of odd order. We can for example use the real representation ring to caluclate $$ KO^0(BG) = \mathbb{Z}[[t]]/(tf_n(t)) $$ with $f_n(t)$ being some complicated polynomial, but is it necessarily true that $$ KO^*(BG) = \pi_*(KO)[[t]]/(tf_n(t)) $$ holds?


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