Finite matrix groups over nilpotent group rings
Let \(\mathbb{Z} H\) be the integral group ring of a group \(H\). The authors study the group \(\text{SGL}_n(\mathbb{Z} H)\) of matrices of augmentation one over the group ring \(\mathbb{Z} H\), and relate the torsion of \(\text{SGL}_n(\mathbb{Z} H)\) to the torsion of \(H\). The authors obtain the following results: Let \(\langle 1\rangle=G_0<G_1<\cdots<G_k=G\triangleleft H\) be a subnormal series of \(H\) and \(K\) be the kernel of the natural homomorphism \(\text{GL}_n(\mathbb{Z} H)\to\text{GL}_n(\mathbb{Z}(H/G))\). Then: 1. if the factor groups \(G_i/G_{i-1}\), \(1\leq i\leq k\) are torsion free abelian, then \(K\) is torsion free; 2. if \(G\) is finite nilpotent, then \(K\) has \(p\)-torsion only for the primes \(p\) dividing the order of \(G\). In particular, if \(H\) is a torsion free nilpotent group, then \(\text{SGL}_n(\mathbb{Z} H)\) is torsion free. Let \(H\) be a polycyclic-by-finite group. If \(H\) has no \(p\)-torsion, then also the matrix groups \(\text{SGL}_n(\mathbb{Z} G)\) have no \(p\)-torsion and in case when \(H\) has only \(p\)-torsion, then every finite subgroup of \(\text{SGL}_n(\mathbb{Z} H)\) is a \(p\)-group. Let \(H\) be a nilpotent group. Then all finite subgroups of \(\text{SGL}_n(\mathbb{Z} G)\) are also nilpotent and every matrix \(X\) of prime power order can be stably diagonalized. The authors apply these results to study finite subgroups \(U\) of \(\text{SGL}_n(\mathbb{Z} G)\). The subgroup has an embedding \(\psi:U\to D(H)=\{\text{diag}(g_1,g_2,\dots,g_n)\}\), where the \(g_i\) are elements of finite order in \(H\). When \(n=1\) we obtain that every finite subgroup of the group of normalized units \(V(\mathbb{Z} H)\) is isomorphic to a subgroup of \(H\). Let \(H\) be a finitely generated nilpotent group and \(t(H)\) the torsion part of \(H\). From \(\mathbb{Z} G\cong\mathbb{Z} H\) it follows that \(G\) is also a finitely generated nilpotent group and \(t(H)\cong t(G)\) and \(G/t(G)\cong H/t(H)\).
scientific article; zbMATH DE number 883774