ISOLATORS IN SOLUBLE GROUPS OF FINITE RANK ... - Project Euclid
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ISOLATORS IN SOLUBLE GROUPS OF FINITE RANK ... - Project Euclid
Feb 2, 1983 - In his Edmonton lectures ([1]) P. Hall proved that locally nilpotent groups ..... P. Hall, The Edmonton Notes on Nilpotent Groups, Queen Mary ...
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 14, Number 2, Spring 1984
ISOLATORS IN SOLUBLE GROUPS OF FINITE RANK
A. H . RHEMTULLA a n d B. A. F. WEHRFRITZ
ABSTRACT. A group G is said to have the isolator property if for every subgroup H of G the set \ZJ[ = [g^G: hn e H for some n ^ 1} is a subgroup. G is said to have the strong isolator property if in addition | VTf: H\ is finite whenever HIÇ\g^GHg is finitely generated. It is well known that locally nilpotent groups have the isolator property, that finitly generated nilpotent groups have the strong isolator property and that the infinite dihedral group D has neither. However D clearly has a subgroup of finite index with the strong isolator property. Our purpose here is to show that for soluble groups and linear groups the isolator property is closely associated with the finiteness of the (Prüfer) rank, while the strong isolator property is closely associated with polycyclicity.
For subgroups H ^ K of a group G let fy~H denote the set {g e K: gneH for some n ^ 1}. If K = G write *J~H for # F . The group G is said to have the isolator property if *J~E is a subgroup of G for every subgroup//of G. If in addition | *J~H : H\ is finite for every//such that H/f)gŒGH8 is finitely generated say that G has the strong isolator property. In his Edmonton lectures ([1]) P. Hall proved that locally nilpotent groups and finitely generated nilpotent groups have the isolator and the strong isolator property respectively. If G0 is the infinite dihedral group and H = < 1 > then trivially *JTi is not a subgroup. However the cyclic subgroup of Go of index 2 clearly does have the strong isolator property. For any class 2 of groups say that a group G is almost a 2*-group if it has a normal 2"-subgroup of finite index. Thus G0 above almost has the strong isolator property. For soluble groups and for linear groups there is a close link between groups that almost have the isolator property and groups with finite (Prüfer) rank. Our main results are as follows. THEOREM A. Let G be a finitely generated soluble group. Then G is polycyclic if and only if G almost has the strong isolator property. THEOREM
B. A torsion-free soluble group of finite rank almost has the
isolator property. If G is a finitely generated, nilpotent-by-polycyclic-byfinite group with the isolator property, then G has finite rank. THEOREM C. Let G be a finitely generated linear group. Then G almost has the isolator property if and only if G is almost soluble-offinite-rank. THEOREM A can be used to give a short proof of the following known result ([3]). Suppose G is a group, H a subgroup of G and Q a set of automorphisms of G Let (H, Q) denote the set of all f j ^ H«> as 0 ranges over all the finite subsets of Û. THEOREM D. Let G be a polycyclic group. Then for every H and Û as above the set (H, Q) satisfies the minimal condition.
A subgroup Hof a. group Gis isolated if// = y/~H. Define the relation ~ on the set of subgroups of G by the rule H ~ ^whenever ^/7f=
