This paper continues a programme of searching for a finite set of generators of a subgroup of finite index in the unit group of an integral group ring of a finite group. Let \$G\$ be a finite group. For an anti-automorphism \$\textbackslash{}varphi\$ of \$G\$, which is naturally extended to \$\textbackslash{}Bbb Z [G]\$, put \$\$ \textbackslash{}scr U \_\{\textbackslash{}varphi\}(\textbackslash{}Bbb Z [G])= \textbackslash{}\{u \textbackslash{}in \textbackslash{}scr U (\textbackslash{}Bbb Z [G])| u \textbackslash{},\textbackslash{}varphi (u)=1 \textbackslash{}\}. \$\$ This group is called the group of \$\textbackslash{}varphi\$-unitary units. Let \$\textbackslash{}varphi \_1, \textbackslash{}dots , \textbackslash{}varphi \_n\$ be anti-automorphisms of \$G\$. The author defines \$\$ \textbackslash{}scr U\_\{ \textbackslash{}varphi\_\{1\},\textbackslash{}dots , \textbackslash{}varphi \_\{n\}\}(\textbackslash{}Bbb Z [G])= \textbackslash{}langle\textbackslash{}scr U \_\{\textbackslash{}varphi \_\{i\}\}(\textbackslash{}Bbb Z[G])| i=1, \textbackslash{}dots , n \textbackslash{}rangle. \$\$ The main result of this paper is that if \$G\$ is a nonabelian group of order less than or equal to \$16\$, then the Bass cyclic units, the bicyclic units and \$\textbackslash{}scr U\_\{ \textbackslash{}varphi\_\{1\}, \textbackslash{}varphi \_\{2\}\}(\textbackslash{}Bbb Z [G])\$, where \$\textbackslash{}varphi \_1, \textbackslash{}varphi \_2\$ are suitably chosen anti-automorphisms of \$G\$, generate a subgroup of finite index in \$\textbackslash{}scr U (\textbackslash{}Bbb Z [G])\$.
Dooms, A 2006, 'Unitary Units in Integral Group Rings', Journal of Algebra and Its Applications, vol. 5, pp. 43-52. <http://www.worldscinet.com/cgi-bin/details.cgi?id=pii:S0219498806001569&type=html>
Dooms, A. (2006). Unitary Units in Integral Group Rings. Journal of Algebra and Its Applications, 5, 43-52. http://www.worldscinet.com/cgi-bin/details.cgi?id=pii:S0219498806001569&type=html
@article{77e2a69deca6414cae7ee7b1ec564d43,
title = "Unitary Units in Integral Group Rings",
abstract = "This paper continues a programme of searching for a finite set of generators of a subgroup of finite index in the unit group of an integral group ring of a finite group. Let \$G\$ be a finite group. For an anti-automorphism \$\textbackslash{}varphi\$ of \$G\$, which is naturally extended to \$\textbackslash{}Bbb Z [G]\$, put \$\$ \textbackslash{}scr U \_\{\textbackslash{}varphi\}(\textbackslash{}Bbb Z [G])= \textbackslash{}\{u \textbackslash{}in \textbackslash{}scr U (\textbackslash{}Bbb Z [G])| u \textbackslash{},\textbackslash{}varphi (u)=1 \textbackslash{}\}. \$\$ This group is called the group of \$\textbackslash{}varphi\$-unitary units. Let \$\textbackslash{}varphi \_1, \textbackslash{}dots , \textbackslash{}varphi \_n\$ be anti-automorphisms of \$G\$. The author defines \$\$ \textbackslash{}scr U\_\{ \textbackslash{}varphi\_\{1\},\textbackslash{}dots , \textbackslash{}varphi \_\{n\}\}(\textbackslash{}Bbb Z [G])= \textbackslash{}langle\textbackslash{}scr U \_\{\textbackslash{}varphi \_\{i\}\}(\textbackslash{}Bbb Z[G])| i=1, \textbackslash{}dots , n \textbackslash{}rangle. \$\$ The main result of this paper is that if \$G\$ is a nonabelian group of order less than or equal to \$16\$, then the Bass cyclic units, the bicyclic units and \$\textbackslash{}scr U\_\{ \textbackslash{}varphi\_\{1\}, \textbackslash{}varphi \_\{2\}\}(\textbackslash{}Bbb Z [G])\$, where \$\textbackslash{}varphi \_1, \textbackslash{}varphi \_2\$ are suitably chosen anti-automorphisms of \$G\$, generate a subgroup of finite index in \$\textbackslash{}scr U (\textbackslash{}Bbb Z [G])\$.",
keywords = "unitary, units",
author = "Ann Dooms",
year = "2006",
month = feb,
day = "1",
language = "English",
volume = "5",
pages = "43--52",
journal = "Journal of Algebra and Its Applications",
issn = "0219-4988",
publisher = "World Scientific Publishing Co. Pte Ltd",
}