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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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