A characterization of the unit group in $\mathbb{Z}[T \times C_2]$

Tevfik Bilgin; Omer Kusmus; Richard M. Low

doi:10.4134/BKMS.b150526

Bulletin of the
Korean Mathematical Society BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2016; 53(4): 1105-1112

Printed July 31, 2016

https://doi.org/10.4134/BKMS.b150526

A characterization of the unit group in $\mathbb{Z}[T \times C_2]$

Tevfik Bilgin, Omer Kusmus, and Richard M. Low

Fatih University, Y\"{u}z\"{u}nc\"{u} Y$\textit{\i}$l University, San Jose State University

Abstract

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Abstract

Describing the group of units $U(\mathbb{Z}G)$ of the integral group ring $\mathbb{Z}G$, for a finite group $G$, is a classical and open problem. In this note, we show that $U_1(\mathbb{Z}[T \times C_2]) \cong [F_{97} \rtimes F_5] \rtimes [T \times C_2]$, where $T = \langle a, b: a^6 = 1, a^3 = b^2, ba = a^5b \rangle$ and $F_{97}$, $F_5$ are free groups of ranks 97 and 5, respectively.

Keywords: integral group ring, unit problem

MSC numbers: Primary 16S34