2012/03/23 by Jespers, Eric, Olteanu, Gabriela, del Río, Ángel +1
#Commutative Algebra (math.AC) #FOS: Mathematics #Group Theory (math.GR) #K-Theory and Homology (math.KT) #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.1203.5232
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 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 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 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)'.