THE REPRESENTATION TYPE OF THE CHARACTER RING

Let R(G) be the character ring of a finite group G. We consider the question whether the representation type of R(G) is finite or infinite. We show that if R(G) is representation-finite, then exp(G) is cube-free and the Sylow subgroups of G are cyclic, elementary-abelian, or nonabelian of order 8. Moreover, we give further necessary as well as some sufficient conditions on the structure of G for the finiteness of the representation type of R(G).