# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Bass, H.: The Dirichlet unit Theorem and Whitehead groups of finite groups. Topology, 4 (1966), 391–410 · Zbl 0166.02401  Bovdi, A.A.: Unitary of the multiplicative group of an integral group ring. Math. USSR Sbornik 47 (2) (1984), 377–389 · Zbl 0527.16004  Bovdi, A.A.: On the construction of integral group rings with trivial elements of finite order. Sibirsk. Math. Zh. 21 (4) (1980), 28–37  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  Bovdi, A.A.: The multiplicative group of an integral group ring. Uzhgorod, 1987 · Zbl 0688.16007  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  Hoechsmann, K. and Sehgal, S.K.: On a Theorem of Bovdi. Publ. Math. Debrecen, to appear 1992  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)  Sehgal, S.K.: Topics in group rings, M. Dekker, New York 1976 · Zbl 0345.20006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.