The effect of an involution in $\widetilde K^0(ZG)$

The involution in the Grothendieck group of the group ring of a finite cyclic group of prime order $p$, induced by the transition to the contragredient module is identical to complex conjugation followed by the automorphism $x\to x^{-1}$ in the ideal class group of the cyclotomic field of order $p$.