Hostname: page-component-77c89778f8-swr86 Total loading time: 0 Render date: 2024-07-16T18:30:19.306Z Has data issue: false hasContentIssue false
  • English
  • Français

Generating groups of nilpotent varieties

Published online by Cambridge University Press:  17 April 2009

M. R. Vaughan-Lee
M. R. Vaughan-Lee
Affiliation:
Vanderbilt University, Nashville, Tennessee, USA.
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.

If is a variety of groups which are nilpotent of class c then is generated by its free group of rank c. It is proved that under certain general conditions cannot be generated by its free group of rank c - 2, and that under certain other conditions is generated by its free group of rank c - 1. It follows from these results that if is the variety of all groups which are nilpotent of class c, then the least value of k such that the free group of of rank k generates is c - 1. This extends known results of L.G. Kovács, M.F. Newman, P.P. Pentony (1969) and F. Levin (1970).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 3 , Issue 2 , October 1970 , pp. 145 - 154
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Neumann, Hanna, Varieties of groups (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 37, Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
[2]Baumslag, Gilbert, Neumann, B.H., Neumann, Hanna and Neumann, Peter M., “On varieties generated by a finitely generated group”, Math. Z. 86 (1964), 93–122.CrossRefGoogle Scholar
[3]Kovács, L.G., Newman, M.F. and Pentony, P.P., “Generating groups of nilpotent varieties”, Bull. Amer. Math. Soc. 74 (1968), 968–971.Google Scholar
[4]Levin, Frank, “Generating groups for nilpotent varieties”, J. Austral. Math. Soc. 11 (1970), 28–32.CrossRefGoogle Scholar
[5]Hall, Marshall Jr, The theory of groups (Macmillan, New York, 1959).Google Scholar
[6]Ward, M.A., “Basic commutators”, Philos. Trans. Roy. Soc. London Ser. A 264 (1969), 343–412.Google Scholar