A generalization of a theorem of Minkowski (Q2759132)

Let \(S\) be a ring and \(I\) an ideal of \(S\). Denote by \(R_I\) the natural map \(\text{GL}_n(S)\to\text{GL}_n(S/I)\) and by \(\text{GL}_n(S,I)\) its kernel. For a subgroup \(G\) of \(\text{GL}_n(S)\), let \(G(I)\) be the kernel of \(R_I\) restricted to \(G\). A theorem of Minkowski says that if \(S\) is the ring of integers or the ring of 2-adic integers, then any finite subgroup of \(\text{GL}_n(S,2S)\) is conjugate in \(\text{GL}_n(S)\) to diagonal matrices.NEWLINENEWLINENEWLINEThe author generalizes the above result. More precisely, suppose either \(K\) is a finite Galois extension of \(\mathbb{Q}\) which is unramified at all finite primes except \(p\) and such that there is a unique prime ideal \(\gamma\) in \(K\) over \(p\), or \(K\) is a finite Galois extension of \(\mathbb{Q}_p\) contained in \(\mathbb{Q}_p(\mu_{p^\infty})\) and \(\gamma\) is the maximal ideal of \(O_K\), where \(\mu_{p^\infty}\) is the group of roots of unity of \(p\)-power order and \(O_K\) is the ring of integers of \(K\). Let \(\Gamma\) be the Galois group of \(K\) and let \(G\subseteq\text{GL}_n(O_K)\) be a finite \(\Gamma\)-stable subgroup. Then \(G(\Gamma)\) is conjugate to diagonal matrices by a matrix in \(\text{GL}_n(\mathbb{Z})\) or \(\text{GL}_n(\mathbb{Z}_p)\).