On the Semisimplicity of Integral Representation Rings by Janice Zemanek

The integral representation algebra A(RG) is C ®z a(RG). When does a(RG) contain nontrivial nilpotent elements? Let | G\ = pn, where p\n, p prime. Denote by Zp the £-adic valuation ring in Q, and by Zp* its completion. Reiner has shown (i) If a = l , then A(ZPG) and A(Z*G) have no nonzero nilpotent elements (see [ l ] ) . (ii) If ce^2, and G has an element of order p, then both A(ZPG) and A{Z*G) contain nonzero nilpotent elements (see [2 j). We have been able to settle the open case as to what happens when G has a (p, £)-subgroup. Our main result is