The representation type of the character ring

Abstract

Let R(G) be the character ring of a nite group G. We consider

the question whether the representation type of R(G) is nite or innite. We

show that if R(G) is representation-nite, 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 sucient conditions on

the structure of G for the niteness of the representation type of R(G).

Keywords

• Character ring,representation type,almost 2-transitive groups of cube-free order

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