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 $\varphi$ of $G$, which is naturally extended to $\Bbb Z [G]$, put $$ \scr U _{\varphi}(\Bbb Z [G])= \{u \in \scr U (\Bbb Z [G])| u \,\varphi (u)=1 \}. $$ This group is called the group of $\varphi$-unitary units. Let $\varphi _1, \dots , \varphi _n$ be anti-automorphisms of $G$. The author defines $$ \scr U_{ \varphi_{1},\dots , \varphi _{n}}(\Bbb Z [G])= \langle\scr U _{\varphi _{i}}(\Bbb Z[G])| i=1, \dots , n \rangle. $$ 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 $\scr U_{ \varphi_{1}, \varphi _{2}}(\Bbb Z [G])$, where $\varphi _1, \varphi _2$ are suitably chosen anti-automorphisms of $G$, generate a subgroup of finite index in $\scr U (\Bbb Z [G])$.
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 $\varphi$ of $G$, which is naturally extended to $\Bbb Z [G]$, put $$ \scr U _{\varphi}(\Bbb Z [G])= \{u \in \scr U (\Bbb Z [G])| u \,\varphi (u)=1 \}. $$ This group is called the group of $\varphi$-unitary units. Let $\varphi _1, \dots , \varphi _n$ be anti-automorphisms of $G$. The author defines $$ \scr U_{ \varphi_{1},\dots , \varphi _{n}}(\Bbb Z [G])= \langle\scr U _{\varphi _{i}}(\Bbb Z[G])| i=1, \dots , n \rangle. $$ 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 $\scr U_{ \varphi_{1}, \varphi _{2}}(\Bbb Z [G])$, where $\varphi _1, \varphi _2$ are suitably chosen anti-automorphisms of $G$, generate a subgroup of finite index in $\scr U (\Bbb Z [G])$.",
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",
}