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Unitary subgroup of integral group rings. (English) Zbl 0779.16014
Let \(\mathbb{Z} G\) be the integral group ring of an arbitrary group \(G\) and \(U(\mathbb{Z} G)\) be the group of units of \(\mathbb{Z} G\). Let \(f:G\to U(\mathbb{Z})\) be an orientation homomorphism of the group \(G\), and for each \(x = \sum_{g\in G}a_ gg\) in \(\mathbb{Z} G\) set \(x^ f = \sum_{g\in G}a_ gf(g)g^{-1}\). The collection of elements \(U_ f(\mathbb{Z} G) \equiv \{u \in U(\mathbb{Z} G)\mid u^{-1} = u^ f\text{ or }u^{-1} = -u^ f\}\) forms a subgroup of \(U(\mathbb{Z} G)\). This subgroup has been investigated by several authors. If \(U_ f(\mathbb{Z} G)=U(\mathbb{Z} G)\), then \(U(\mathbb{Z} G)\) is called \(f\)-unitary. By the first author [in Mat. Sb., Nov. Ser. 119, No. 3, 387-400 (1982; Zbl 0511.16009)] necessary conditions for \(U(\mathbb{Z} G)\) to be \(f\)-unitary were given and many of them were shown to be sufficient. In the present paper, the authors’ main result is on the problem of normality of \(U_ f(\mathbb{Z} G)\) in \(U(\mathbb{Z} G)\). They give necessary conditions for normality. These conditions are sufficient in many cases and they imply easily the necessary conditions for \(U(\mathbb{Z} G)\) to be \(f\)-unitary. One of the sufficiency cases in the above mentioned paper for \(f\)-unitarity of \(U(\mathbb{Z} G)\) is improved while another case still remains open.
Reviewer: T.Akasaki (Irvine)

MSC:
16U60 Units, groups of units (associative rings and algebras)
20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
16S34 Group rings
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