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Central Units of integral group rings

Journal Contribution - Journal Article

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 $\Z G$ 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 $\mathcal{Z} (\U (\Z G))$ of the unit group $\U (\Z G)$ in case $G$ is a finite nilpotent group.
Next, we give a new construction of units that generate a subgroup of finite index in $\mathcal{Z}(\U(\Z G))$ for all finite strongly monomial groups
$G$. We call these units generalized Bass units. Finally, we show that the commutator group $\U(\Z G)/\U(\Z G)'$ and $\mathcal{Z}(\U(\Z G))$ have the
same rank if $G$ is a finite group such that $\Q G$ 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 $\Q$. This allows us to prove that in this case the natural images of the Bass units of $\Z G$ generate a subgroup of finite index in
$\U(\Z G)/\U(\Z G)'$.
Journal: Proc. Amer. Math. Soc.
ISSN: 0002-9939
Volume: 142
Pages: 2193-2209
Publication year:2014
Keywords:group ring, central unit, generators