Central units of integral group rings

Abstract:

We give an explicit description for a basis of a subgroup of finite index in the group of central units of the integral group ring $\mathbb {Z} G$ of a finite abelian-by-supersolvable group such that every cyclic subgroup of order not a divisor of 4 or 6 is subnormal in $G$. The basis elements turn out to be a natural product of conjugates of Bass units. This extends and generalizes a result of Jespers, Parmenter and Sehgal showing that the Bass units generate a subgroup of finite index in the center $\mathcal {Z} (\mathcal {U} (\mathbb {Z} G))$ of the unit group $\mathcal {U} (\mathbb {Z} G)$ in case $G$ is a finite nilpotent group. Next, we give a new construction of units that generate a subgroup of finite index in $\mathcal {Z}(\mathcal {U}(\mathbb {Z} G))$ for all finite strongly monomial groups $G$. We call these units generalized Bass units. Finally, we show that the commutator group $\mathcal {U}(\mathbb {Z} G)/\mathcal {U}(\mathbb {Z} G)’$ and $\mathcal {Z}(\mathcal {U}(\mathbb {Z} G))$ have the same rank if $G$ is a finite group such that $\mathbb {Q} G$ has no epimorphic image which is either a non-commutative division algebra other than a totally definite quaternion algebra or a two-by-two matrix algebra over a division algebra with center either the rationals or a quadratic imaginary extension of $\mathbb {Q}$. This allows us to prove that in this case the natural images of the Bass units of $\mathbb {Z} G$ generate a subgroup of finite index in $\mathcal {U}(\mathbb {Z} G)/\mathcal {U}(\mathbb {Z} G)’$.