Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-29T00:00:26.114Z Has data issue: false hasContentIssue false
  • English
  • Français

A Property of Groups with No Central Factors

Published online by Cambridge University Press:  20 November 2018

A. H. Rhemtulla
A. H. Rhemtulla*
Affiliation:
The University of Alberta Edmonton, Alberta
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let C1 denote the class of all groups with no non-trivial central factors. We prove the following theorem

There exist non-trivial locally solvable C1 groups; but there is no non-trivial locally k-step polynilpotent C1 group for any integer k.

It is well known that a minimal normal subgroup of a locally solvable group is abelian. Thus no non-trivial locally solvable group can be pluperfect - the class of all perfect groups in which every subnormal subgroup is also perfect.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 4 , August 1969 , pp. 467 - 470
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Dark, R. and Rhemtulla, A. H., On RO-closed classes, and finitely generated groups. Canad. J. Math, (to appear).Google Scholar
2. Hall, P., Wreath powers and characteristically simple groups. Proc. Camb. Phil. Soc. 58 (1962) 170–184.Google Scholar
3. McLain, D. H., A characteristically simple group. Proc. Camb. Phil. Soc. 50 (1954) 641–642.Google Scholar
4. McLain, D.H., Finiteness conditions in locally soluble groups. Journal London Math. Soc. 34 (1959) 101–107.Google Scholar
5. Scott, W.R., Group theory. (Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1964.)Google Scholar
6. Taunt, D., On A-Groups. Proc. Camb. Phil. Soc. 45 (1949) 24–42.Google Scholar