Polycyclic groups and topological groups. (English) Zbl 0705.20032
Group theory, Proc. 2nd Int. Conf., Bressanone/Italy 1989, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 23, 63-71 (1990).
[For the entire collection see Zbl 0695.00013.]
P. Hall has shown that, with respect to a Mal’cev basis for a torsion- free nilpotent group N, the group operations can be represented by polynomial mappings. As a consequence, it is possible to associate with N an algebraic group which provides a powerful tool for the study of N. See, for example, the book of D. Segal [Polycyclic Groups (Cambridge Tracts Math. 82, 1983; Zbl 0516.20001)]. Here similar constructions for a class of polycyclic groups are described. Call a polycyclic group weakly splittable if it is the extension of a torsion- free Fitting subgroup by a free abelian group. Then every polycyclic group \(\Gamma\) has a subgroup of finite index which is weakly splittable.
The group \(\Gamma\) can be embedded in a larger group \(\Gamma^ Q\) obtained by embedding the Fitting subgroup into its Mal’cev completion and within this group an analogue of a Mal’cev basis is defined. The elements of \(\Gamma^ Q\) are then in 1-1 correspondence with \((n+m)\)- tuples of numbers, the first n being rational and the last m being integral. It is shown that the binary function on these \((n+m)\)-tuples which is associated with multiplication in the group is given by polynomials in the entries of the \((n+m)\)-tuples together with entries of the form \(w^ r\) where w is an algebraic number and r is one of the integral entries in the last m places of the \((n+m)\)-tuple.
It is then shown how to use this to obtain analytic groups \(\Gamma^ C\) associated with \(\Gamma\). These groups are not in general algebraic, but a variation on the construction yields an algebraic group \(\Gamma^ A\) associated with \(\Gamma\). Although there are various possibilities for the analytic groups \(\Gamma^ C\) associated with a particular \(\Gamma\), the final algebraic group \(\Gamma^ A\) is uniquely determined by \(\Gamma\). This canonical algebraic group is shown to coincide with one defined by S. Donkin [in J. Reine Angew. Math. 326, 104-123 (1981; Zbl 0453.20028)] which was also obtained in a different way by A. Magid [in J. Algebra 74, 149-158 (1982; Zbl 0505.20007)]. The intermediate analytic groups \(\Gamma^ C\) are also shown to include analytic groups considered by Magid.
The paper gives a brief discussion of the major results together with some examples. No proofs are given.
P. Hall has shown that, with respect to a Mal’cev basis for a torsion- free nilpotent group N, the group operations can be represented by polynomial mappings. As a consequence, it is possible to associate with N an algebraic group which provides a powerful tool for the study of N. See, for example, the book of D. Segal [Polycyclic Groups (Cambridge Tracts Math. 82, 1983; Zbl 0516.20001)]. Here similar constructions for a class of polycyclic groups are described. Call a polycyclic group weakly splittable if it is the extension of a torsion- free Fitting subgroup by a free abelian group. Then every polycyclic group \(\Gamma\) has a subgroup of finite index which is weakly splittable.
The group \(\Gamma\) can be embedded in a larger group \(\Gamma^ Q\) obtained by embedding the Fitting subgroup into its Mal’cev completion and within this group an analogue of a Mal’cev basis is defined. The elements of \(\Gamma^ Q\) are then in 1-1 correspondence with \((n+m)\)- tuples of numbers, the first n being rational and the last m being integral. It is shown that the binary function on these \((n+m)\)-tuples which is associated with multiplication in the group is given by polynomials in the entries of the \((n+m)\)-tuples together with entries of the form \(w^ r\) where w is an algebraic number and r is one of the integral entries in the last m places of the \((n+m)\)-tuple.
It is then shown how to use this to obtain analytic groups \(\Gamma^ C\) associated with \(\Gamma\). These groups are not in general algebraic, but a variation on the construction yields an algebraic group \(\Gamma^ A\) associated with \(\Gamma\). Although there are various possibilities for the analytic groups \(\Gamma^ C\) associated with a particular \(\Gamma\), the final algebraic group \(\Gamma^ A\) is uniquely determined by \(\Gamma\). This canonical algebraic group is shown to coincide with one defined by S. Donkin [in J. Reine Angew. Math. 326, 104-123 (1981; Zbl 0453.20028)] which was also obtained in a different way by A. Magid [in J. Algebra 74, 149-158 (1982; Zbl 0505.20007)]. The intermediate analytic groups \(\Gamma^ C\) are also shown to include analytic groups considered by Magid.
The paper gives a brief discussion of the major results together with some examples. No proofs are given.
Reviewer: J.R.J.Groves
MSC:
| 20F16 | Solvable groups, supersolvable groups |
| 20E22 | Extensions, wreath products, and other compositions of groups |
| 20G30 | Linear algebraic groups over global fields and their integers |
| 20E18 | Limits, profinite groups |
| 20E07 | Subgroup theorems; subgroup growth |
