Unitary subgroup of integral group rings.

*(English)*Zbl 0779.16014Let \(\mathbb{Z} G\) 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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\textit{A. A. Bovdi} and \textit{S. K. Sehgal}, Manuscr. Math. 76, No. 2, 213--222 (1992; Zbl 0779.16014)

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##### References:

[1] | Bass, H.: The Dirichlet unit Theorem and Whitehead groups of finite groups. Topology, 4 (1966), 391–410 · Zbl 0166.02401 |

[2] | Bovdi, A.A.: Unitary of the multiplicative group of an integral group ring. Math. USSR Sbornik 47 (2) (1984), 377–389 · Zbl 0527.16004 |

[3] | Bovdi, A.A.: On the construction of integral group rings with trivial elements of finite order. Sibirsk. Math. Zh. 21 (4) (1980), 28–37 |

[4] | Bovdi, A.A.: Unitary subgroup of the multiplicative group of integral group ring of a cyclic group. Math. Zametki 41 (4) (1987), 469–474 · Zbl 0632.16010 |

[5] | Bovdi, A.A.: The multiplicative group of an integral group ring. Uzhgorod, 1987 · Zbl 0688.16007 |

[6] | Cliff, G.H. and Sehgal, S.K.: Groups which are normal in the unit group of their group rings. Arch. Math. 33 (6) (1979), 528–537 · Zbl 0418.20004 |

[7] | Hoechsmann, K. and Sehgal, S.K.: On a Theorem of Bovdi. Publ. Math. Debrecen, to appear 1992 |

[8] | Novikov, S.P.: Algebraic construction and properties of Hermitian analogues ofK-theory over rings with involution from the viewpoint of Hamiltonian formalism. Applications to differential topology and the theory of characteristic classes. II, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 475–500; English transl. in Math. USSSR Izv. 4 (1970) |

[9] | Sehgal, S.K.: Topics in group rings, M. Dekker, New York 1976 · Zbl 0345.20006 |

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