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On Torsion-Free Groups of Finite Rank

Published online by Cambridge University Press:  20 November 2018

David Meier  and
Akbar Rhemtulla
David Meier
Affiliation:
University of Alberta, Edmonton, Alberta
Akbar Rhemtulla
Affiliation:
University of Alberta, Edmonton, Alberta
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This paper deals with two conditions which, when stated, appear similar, but when applied to finitely generated solvable groups have very different effect. We first establish the notation before stating these conditions and their implications. If H is a subgroup of a group G, let denote the set

We say G has the isolator property if is a subgroup for all H ≦ G. Groups possessing the isolator property were discussed in [2]. If we define the relation ∼ on the set of subgroups of a given group G by the rule H ∼ K if and only if , then ∼ is an equivalence relation and every equivalence class has a maximal element which may not be unique. If , we call H an isolated subgroup of G.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 6 , 01 December 1984 , pp. 1067 - 1080
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Hall, P., Some sufficient conditions for a group to be nilpotent, Illinois J. Math. 2 (1958), 787–801.Google Scholar
2. Rhemtulla, A. H. and Wehrfritz, B. A. F., Isolators in soluble groups of finite rank, Rocky Mountain J. Math. 14 (1984), 415–422.Google Scholar
3. Rhemtulla, A. H., Weiss, A. and Yousif, M., Solvable groups with π-isolators, Proc. Amer. Math Soc. 90 (1984), 173–178.Google Scholar
4. Robinson, D. J. S., Finiteness conditions and generalized soluble groups, Parts 1 and 2 (Springer Verlag, 1972).CrossRefGoogle Scholar