Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T02:31:33.040Z Has data issue: false hasContentIssue false
  • English
  • Français

Characteristic subgroups of relatively free groups

Published online by Cambridge University Press:  17 April 2009

Roger M. Bryant
Roger M. Bryant
Affiliation:
Department of Mathematics, University of Manchester Institute of Science and Technology, Manchester M60 1QD, United Kingdom
Rights & Permissions [Opens in a new window]

Abstract

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.

A simple new proof is given of a result of Vaughan-Lee which implies that if G is a relatively free nilpotent group of finite rank k and nilpotency class c with c < k then the characteristic subgroups of G are all fully invariant. It is proved that the condition c < k can be weakened to c < k + p − 2 when G has p–power exponent for some prime p. On the other hand it is shown that for each prime p there is a 2-generator relatively free p-group G which is nilpotent of class 2p such that the centre of G is not fully invariant.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 46 , Issue 3 , December 1992 , pp. 497 - 507
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Bryant, R.M. and Kovács, L.G., ‘Lie representations and groups of prime power order’, J. London Math. Sac. (2) 17 (1978), 415–421.CrossRefGoogle Scholar
[2]Huppert, B. and Blackburn, N., Finite groups II(Springer-Verlag, Berlin, Heidelberg, New York, 1982).CrossRefGoogle Scholar
[3]Magnus, W., Karrass, A. and Solitar, D., Combinatorial group theory (Interscience, New York, 1966).Google Scholar
[4]Neumann, B.H. and Neumann, H., ‘Zwei Klassen charakteristischer Untergruppen und ihre Faktorgruppen’, Math. Nachr. 4 (1951), 106–125.Google Scholar
[5]Neumann, H., Varieties of groups (Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
[6]Neumann, P.M., ‘A note on formations of finite nilpotent groups’, Bull. London Math. Soc. 2 (1970), 91.CrossRefGoogle Scholar
[7]Vaughan-Lee, M.R., ‘Characteristic subgroups of free groups’, Bull. London Math. Soc. 2 (1970), 87–90.CrossRefGoogle Scholar