On the torsion units in integral group rings of dihedral \(2\)-groups
Let \(D_{2^n}\) be the dihedral group of order \(2^{n+1}\). The number of involutions in the group \(V(\mathbb{Z} D_{2^n})\) of units of augmentation 1 in the integral group ring \(\mathbb{Z} D_{2^n}\) is found to be \(1+2^n \cdot \prod^n_{k=1} h^+_{2^k}\), where \(h^+_{2^k}\) is the class number of the maximal real subfield of \(\mathbb{Q}(\zeta_{2^k})\), \(\zeta_{2^k}\) a primitive \(2^k\)-root of unity. The number of these involutions that belong to some group basis is \(1+2^n\). Hence, each involution belongs to a group basis if and only if each \(h^+_{2^k}=1\). Number theorists conjecture that this is indeed true. The interest in this is motivated by the result of Hughes and Pearson that \(\mathbb{Z} S_3\) has an involution of augmentation 1 that is not contained in any group basis. The proof, which uses class groups and Mayer-Vietoris sequences, also yields a description of the automorphism group of \(\mathbb{Z} D_{2^n}\). Finally, it is shown that the integral group ring of a semidihedral group of order at least 32 has an augmented involution that is not in any group basis.
scientific article; zbMATH DE number 783914