Zassenhaus conjecture and infinite nilpotent groups (Q1923952)

Let \(G\) be a group and let \(\mathbb{Z} G\) be its integral group ring. Let \(C_\infty\) be the infinite cyclic group. Then, \(C_\infty\) operates on the rank two free abelian group via the matrix \(\left(\begin{smallmatrix} 1 & 1\\ 0 & 1\end{smallmatrix}\right)\) and the semidirect product of \(C_\infty\) with this module is \(H\). Let \(G\) be the direct product of \(H\) with \(D_8\), the dihedral group of order 8. The authors give explicitly a unit \(u\) of order 2 in \(\mathbb{Z} H\) which is not conjugate in \(\mathbb{Q} G\) to an element in \(\pm1 G\). For finite nilpotent \(G\) a theorem of Weiss shows that any finite subgroup of the units of \(\mathbb{Z} G\) is conjugate in the units of \(\mathbb{Q} G\) to a subgroup of \(\langle\pm1,G\rangle\). The above \(G\) is nilpotent non finite. It was a conjecture of Zassenhaus that for any finite \(G\) and any finite subgroup \(H\) of the units of \(\mathbb{Z} G\) such that the order of \(H\) equals the order of \(G\), the group \(H\) is conjugate in the units of \(\mathbb{Q} G\) to a subgroup of \(\langle\pm 1,G\rangle\). Roggenkamp and Scott gave a counterexample in general and proved the conjecture for finite nilpotent groups \(G\). The dihedral group of order 8 has a two dimensional faithful representation in the two dimensional vector space over the rationals. The proof uses the tensor product of this representation with the regular representation of \(\mathbb{Q} H\). In order to prove the main theorem the arithmetic and the ring structure of \(\mathbb{Q} H\) is intensively used.