Rational permutation modules for finite groups

By the Artin Induction theorem,C(G) is a finite abelian group with exponent dividing the order of G. Some work on this sequence has already been done. In [14] and [16], Ritter and Segal proved that C(G) = 0 for G a finite p–group. Serre [17, p. 104] remarked that C(G) / = 0 for G = Z/3 × Q8 (the direct product of a cyclic group of order 3 and a quaternion group of order 8). Berz [2] gave a nice description of P (G) for G metabelian or supersolvable. To describe the result, recall thatR(G) additively is a free abelian group with basis given by the irreducible rational representations ofG. The subgroup P (G) is generated by the induced representations IndG(1H ) = Q[G/H], whereH runs over the subgroups ofG. If aφ denotes the gcd over all H of the numbers 〈φ, IndG(1H )〉, then aφ divides 〈φ, χ〉 whenever χ is a virtual permutation representation. Let αφ = aφ 〈φ,φ〉 . Theorem: (Berz [2]) ForGmetabelian or supersolvable the latticeP (G) ⊆ R(G) has a basis αφ · φ where φ runs over the irreducible rational representations of G.