Unitary Units in Integral Group Rings

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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 $\scr U (\Bbb Z [G])$.
Original languageEnglish
Pages (from-to)43-52
Number of pages10
JournalJournal of Algebra and Its Applications
Volume5
Publication statusPublished - 1 Feb 2006

Keywords

  • unitary
  • units

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