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This article is cited in 7 scientific papers (total in 7 papers)
Unitarity of the multiplicative group of an integral group ring
A. A. Bovdi
Abstract:
A homomorphism $f$ of a group $G$ into the multiplicative group of the ring of integers is called, in algebraic topology, an orientation homomorphism of the group $G$.
If $x=\sum_{g\in G}\alpha_g g$ is an element of the integral group ring $ZG$, we will let $x^f$ denote the element $\sum_{g\in G}\alpha_g f(g)g^{-1}$. An element $x$ of the multiplicative group $U(ZG)$ is called $f$-unitary if the inverse $x^{-1}$ coincides with $x^f$ or $x^{-f}$. The collection of all $f$-unitary elements of the group $U(ZG)$ form a subgroup $U_f(ZG)$. If $U_f(ZG)=U(ZG)$, the group $U(ZG)$ is said to be $f$-unitary.
Our study of the group $~U_f(ZG)$ is motivated by its appearance in algebraic topology, and was suggested by S. P. Novikov.
The main result of this article consists of necessary conditions, given in terms of the kernel $\operatorname{Ker}f$ and an element $b$ such that $G=\langle\operatorname{Ker}f,b\rangle$, for the group $U(ZG)$ to be $f$-unitary. We also consider to what extent these conditions are sufficient.
Bibliography: 3 titles.
Received: 07.04.1982
Citation:
A. A. Bovdi, “Unitarity of the multiplicative group of an integral group ring”, Mat. Sb. (N.S.), 119(161):3(11) (1982), 387–400; Math. USSR-Sb., 47:2 (1984), 377–389
Linking options:
https://www.mathnet.ru/eng/sm2891https://doi.org/10.1070/SM1984v047n02ABEH002649 https://www.mathnet.ru/eng/sm/v161/i3/p387
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Abstract page: | 322 | Russian version PDF: | 97 | English version PDF: | 7 | References: | 35 | First page: | 1 |
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