On the torsion-free crystallographic groups with indecomposable cyclic point \(p\)-group (Q2730546)

The authors study the torsion-free crystallographic groups with indecomposable cyclic point \(p\)-group. The main results are the following. Let \(G_s\) be a cyclic \(p\)-group of order \(p^s\). Then the problem of classification of torsion-free crystallographic groups with point group isomorphic to the group \(G_s\) is wild, when \(s>4\). Denote by \(m'(G_s)\) the least dimension for which there exist indecomposable crystallographic groups with point group isomorphic to the group \(G_s\). For the cyclic group \(G_1\) of order \(p\) there do not exist indecomposable torsion-free crystallographic groups with point group isomorphic to the group \(G_1\). Let \(s>1\), then \(m'(G_s)=p^s-p^{s-1}+p\). In dimension \(m'(G_s)\) there exist exactly \(p-1\) indecomposable torsion-free crystallographic groups with point groups isomorphic to the group \(G_s\) and pairwise nonconjugate in the group \(\text{GL}(m'(G_s),\mathbb{Z}_p)\).