On the normalizer problem for integral group rings of torsion groups
Given a group \(G\), let \({\mathcal U}(\mathbb{Z} G)\) be the group of units of its integral group ring \(\mathbb{Z} G\). The normalizer \(N_{\mathcal U}(G)\) of \(G\) in \({\mathcal U}(\mathbb{Z} G)\) clearly contains \(G\) and the centre \({\mathcal Z}={\mathcal Z}({\mathcal U}(\mathbb{Z} G))\) of \({\mathcal U}(\mathbb{Z} G)\). The normalizer problem asks for a description of the quotient \(N_{\mathcal U}(G)/G{\mathcal Z}\). The group \(G\) is said to have the normalizer property if \(N_{\mathcal U}(G)=G{\mathcal Z}\), or equivalently if \(\Aut_{\mathcal U}(G)=\text{Inn}(G)\), where \(\Aut_{\mathcal U}(G)\) is the group of automorphisms of \(G\) induced by the elements of \({\mathcal U}(\mathbb{Z} G)\). Extending a result of \textit{D. B. Coleman} [Proc. Am. Math. Soc. 15, 511-514 (1964; Zbl 0132.27501)], the author proves that locally finite nilpotent groups have the normalizer property. The second main result proved in this paper is that a torsion group \(G\) has the normalizer property if for a maximal Sylow 2-subgroup \(P\) of \(G\), every involution \(\varphi\in\Aut_{\mathcal U}(G)\) which keeps \(P\) pointwise fixed is inner; this extends a result of \textit{S. Jackowski} and \textit{Z. Marciniak} [J. Pure Appl. Algebra 44, 241-250 (1987; Zbl 0624.20024)].
scientific article; zbMATH DE number 1874391