Finite conjugacy in algebras and orders (Q2716990)

Let \(G\) be a group. The authors study the set \(\Delta(G)\) of elements of \(G\) for which their centralizer in \(G\) is a subgroup of finite index in \(G\). In particular, the case of \(G\) being the unit group of a simple algebra is studied. The authors obtain among other results that if \(A\) is in addition finite-dimensional over its centre then \(\Delta(U(A))\) equals the centre of \(U(A)\).NEWLINENEWLINENEWLINEThe results imply similar statements for classical orders in semisimple Artinian algebras. The case of integral group rings \(\mathbb{Z}\Gamma\) of finite groups is studied. Many nice results are obtained such as the following. If the Wedderburn decomposition of the finite group \(\Gamma\) does not contain a skew field, then \(\Delta(U(\mathbb{Z}(\Gamma\times F)))=Z(U(\mathbb{Z}(\Gamma\times F)))\) for any free Abelian group \(F\).