Central units of integral group rings of monomial groups

Abstract

In this paper, it is proved that the group generated by Bass units contains a subgroup of finite index in the group of central units Z ( U ( Z G ) ) \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) of the integral group ring Z G \mathbb {Z}G for a subgroup closed monomial group G G with the property that every cyclic subgroup of order not a divisor of 4 4 or 6 6 is subnormal in G G . If G G is a generalized strongly monomial group, then it is also shown that the group generated by generalized Bass units contains a subgroup of finite index in Z ( U ( Z G ) ) \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) . Furthermore, for a generalized strongly monomial group G G , the rank of Z ( U ( Z G ) ) \mathcal {Z}(\mathcal {U}(\mathbb {Z}G)) is determined. The formula so obtained is in terms of generalized strong Shoda pairs of G G .