Central units of integral group rings of monomial groups

Bakshi, Gurmeet; Kaur, Gurleen

doi:10.1090/proc/15975

Central units of integral group rings of monomial groups
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by Gurmeet K. Bakshi and Gurleen Kaur PDF

Proc. Amer. Math. Soc. 150 (2022), 3357-3368 Request permission

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 $\mathcal {Z}(\mathcal {U}(\mathbb {Z}G))$ of the integral group ring $\mathbb {Z}G$ for a subgroup closed monomial group $G$ with the property that every cyclic subgroup of order not a divisor of $4$ or $6$ is subnormal in $G$. If $G$ 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 $\mathcal {Z}(\mathcal {U}(\mathbb {Z}G))$. Furthermore, for a generalized strongly monomial group $G$, the rank of $\mathcal {Z}(\mathcal {U}(\mathbb {Z}G))$ is determined. The formula so obtained is in terms of generalized strong Shoda pairs of $G$.

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