Metabelian \(p\)-groups and coclass theory.
The paper under review is an important contribution to the program of studying families of finite \(p\)-groups by coclass, and tackles one instance of the major problem of showing that the isomorphism types of finite \(p\)-groups of a fixed coclass can be described in terms of a finite set of data.NEWLINENEWLINE The groups considered here are the metabelian ones, that is, those finite \(p\)-groups in which the derived subgroup is Abelian. The first result shows that for every \(r\), and for every prime \(p\), the isomorphism types of metabelian \(p\)-groups of coclass \(r\) consist of finitely many coclass families, plus finitely many other groups. From this it follows that the graph \(\mathcal M(p,r)\) associated to these groups is virtually periodic, that is, for every coclass tree \(\mathcal M\) in \(\mathcal M(p,r)\), there exist \(d,l\) such that \(\mathcal M_l\cong\mathcal M_{l+d}\). Here the graph \(\mathcal M(p,r)\) has the finite metabelian \(p\)-groups of coclass \(r\) as vertices, and there is an edge from \(G\) to \(H\) if \(H\) has nilpotence class \(c\), and \(H/\gamma_c(H)\cong G\). If \(S\) is an infinite metabelian pro-\(p\)-group of finite coclass \(r\), the coclass tree \(\mathcal M=\mathcal M(S)\) associated to \(S\) is the full subtree of \(\mathcal M(p,r)\) whose vertices are the descendant of the smallest finite quotient of \(S\) of coclass \(r\), which does not occur for some other \(\widehat S\). And for such a coclass tree, one denotes by \(\mathcal M_l\) a pruned version of it, that is, the full subgraph consisting of the groups whose distance to a main line group in \(\mathcal M(p,r)\) is at most \(l\). (We refer to the original paper, which is extremely well written, for further details.)NEWLINENEWLINE From this, one obtains that, for given \(p,r\), with \(p\) prime, metabelian \(p\)-groups of coclass \(r\) can be classified up to isomorphism, that is, their isomorphism classes can be described in terms of a finite number of consistent parametrized presentations.
scientific article; zbMATH DE number 6366944
scientific article