Geodesic
The effect of an involution in $\widetilde K^0(ZG)$
Matematičeskie zametki, Tome 3 (1968) no. 5, pp. 523-527.

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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 $p$. 
@article{MZM_1968_3_5_a3,
     author = {I. Reiner},
     title = {The effect of an involution in $\widetilde K^0(ZG)$},
     journal = {Matemati\v{c}eskie zametki},
     pages = {523--527},
     publisher = {mathdoc},
     volume = {3},
     number = {5},
     year = {1968},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1968_3_5_a3/}
}
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I. Reiner. The effect of an involution in $\widetilde K^0(ZG)$. Matematičeskie zametki, Tome 3 (1968) no. 5, pp. 523-527. http://geodesic.mathdoc.fr/item/MZM_1968_3_5_a3/