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The general polycyclic group. (English) Zbl 0606.20030
The author discusses an approach to the classification of polycyclic groups. This starts from and generalizes the similar approach to the classification of finitely generated torsion-free nilpotent groups discussed [in Group Theory; Essays for Philip Hall, 121-158 (1984; Zbl 0563.20033)] by F. Grunewald and the author. In that paper, the major step was to embed a finitely generated torsion-free nilpotent group into its Mal’cev completion and then rephrase the classification problem in terms of questions concerning algebraic groups and their automorphism groups and arithmetic subgroups.
In this case the approach is similar but necessarily more complicated. The role of Mal’cev completions (that is, of torsion free nilpotent divisible groups of finite rank or D-groups) is played by the set of extensions of D-groups by free abelian groups of finite rank (call these E-groups). It is well known that every polycyclic subgroup has a subgroup of finite index which embeds in such a group.
A general E-group is constructed from a pair of D-groups \(U\leq V\) with U/V abelian together with an embedding into U/V of a suitable free abelian subgroup of the automorphism group of U. Necessary and sufficient conditions for isomorphism of two groups constructed in this way are then given. The problem of classifying E-groups is discussed as is the relation to the original problem of classifying polycyclic groups. Finally, the author notes that these techniques lead to a solution of the isomorphism problem for polycyclic groups.
Reviewer: J.R.J.Groves
MSC:
20F16 Solvable groups, supersolvable groups
20E34 General structure theorems for groups
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
20F18 Nilpotent groups
20E07 Subgroup theorems; subgroup growth
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