Groups with less than n subgroups of index n
Copyright policy )Groups with less than n subgroups of index n
Let \(a_n(G)\) be the number of subgroups of index \(n\) in a group \(G\). The author presents a positive solution to a problem posed by \textit{A. Shalev} [Int. J. Algebra Comput. 7, No. 1, 77-91 (1997; Zbl 0876.20017)] proving that for a residually-finite group the following are equivalent: (1) \(a_n(G)2,\) the pro-\(\{2,p\}\) dihedral group \(G=\langle -1\rangle\ltimes\mathbb{Z}_p\) satisfies \(a_n(G)\leq n\) for all \(n\in\mathbb{N}\), but does not have a central subgroup of finite index; moreover for every positive real number \(\varepsilon\) there exist a prime \(p\) and a non-soluble pro-\(p\) group \(G\) such that \(a_{p^n}(G)<(1+\varepsilon)p^n\) for almost all \(n\in\mathbb{N}\).
- University of Duesseldorf Germany
- Heinrich Heine University Düsseldorf Germany
subgroup growth, numbers of subgroups, Subgroup theorems; subgroup growth, finitely generated residually finite groups, finite cyclic quotients, subgroups of finite index, profinite groups, central cyclic subgroups of finite index, Limits, profinite groups, pro-\(p\) groups, Residual properties and generalizations; residually finite groups
subgroup growth, numbers of subgroups, Subgroup theorems; subgroup growth, finitely generated residually finite groups, finite cyclic quotients, subgroups of finite index, profinite groups, central cyclic subgroups of finite index, Limits, profinite groups, pro-\(p\) groups, Residual properties and generalizations; residually finite groups
selected citations These citations are derived from selected sources.This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).2 popularity This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.Average influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).Average impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.Average
Citations provided by BIP!Popularity provided by BIP!