Pro-\(p\) groups with few normal subgroups.
The authors study pro-\(p\) groups with few normal subgroups in the paper. There are several notions regarding this and the authors study their interrelation as well as connections with existent notions of finite coclass, just infinity, obliquity. A pro-\(p\) group \(G\) has finite coclass if the index of the \(i\)-th term of the lower central series \([G:\gamma_i(G)]\leq p^{i+c-1}\) for some constant \(c\). Put \(\mu_i(G)=\gamma_i(G)\cap\bigcap_{N\triangleleft G,\;N\not\leq\gamma_i(G)}N\). A pro-\(p\) group \(G\) has finite obliquity if \(\sup_i[\gamma_i(G):\mu_i(G)]\) is finite. The authors consider the following conditions on a pair of open normal subgroups \(M\) and \(N\) of a pro-\(p\) group \(G\): (a) There exists a constant \(c\) such that for all \(N,M\) with \(M\not\subset N\) one has \(|N:M\cap N|\leq p^c\). (b) There exists \(c\) such that \(|N:M\cap N|\leq p^c\) for all pairs of subgroups \(M,N\) with the same index in \(G\). (c) For every \(N\) there is a constant \(c\) depending on \(N\) such that \(|N:M\cap N|\leq p^c\) if \(M\not\subset N\). A pro-\(p\) group of finite coclass satisfies the condition (b). The authors prove the following results: 1. Condition (a) is equivalent to \(G\) having finite obliquity. 2. Condition (b) is equivalent to the condition of Constant Normal Subgroup Growth, i.e., to the existence of a constant \(l\) such that \(G\) has at most \(l\) normal subgroups of any given index. 3. Condition (c) is equivalent to \(G\) being just infinite. The second part of the paper introduces the notion of a periodic map on the lattice of subgroups of \(G\). A periodic map on a pro-\(p\) group \(G\) is a map \(\tau\colon M\to G\) from an open normal subgroup \(M\) of \(G\) such that \(\tau(M)\) is an open subgroup of \(G\) and for every open normal subgroup \(H\) of \(G\) contained in \(\tau(M)\), the preimage \(\tau^{-1}(N)\) is an open normal subgroup of \(G\) with \(|G:\tau^{-1}(N)|<|G:N|\). The authors prove that the existence of a periodic map on a just infinite nonabelian pro-\(p\) group \(G\) implies finite obliquity and Constant Normal Subgroup Growth, in fact the normal subgroup growth is periodic (Theorem 50). Furthermore, the authors show that branch groups do not have CNSG (Corollary 60) and so it is natural to restrict the considerations to hereditarily just infinite pro-\(p\) groups. For most of the known examples (Nottingham group (\(p>2\)), Ershov groups, and most \(t\)-linear groups) the authors show the existence of a periodic map and so finite obliquity.
scientific article; zbMATH DE number 5545842
scientific article